BRIEF
According to Euripides, the dowry of Klytaimnestra [Clytaemnestra/ Clytemnestra]1 is a male slave and some other undisclosed items (and/or persons?).
We do not have any such detail(s) about what Helen's dowry is, unless we guess that the entire kingdom of Lakedaimon is this dowry. I'm not convinced that such is necessarily the case. (More on that below.)
From his father Atreus, Agamemnon inherits the kingdom of Argolis, which is already wealthy and powerful at the time of his accession to the throne thereof. He goes on to augment its power through his own agency.
There is no obvious connection between his marriage and his status, except insofar as it appears to be strategically beneficial in terms of political alliance. On top of that, things are yet easier for him in this regard because his brother Menelaus, with whom he is on good terms, has already married into the same family by the time that Agamemnon weds Klytaimnestra.
Apart from gaining a high-class wife, as well as further consolidation of power between Argolis and Lakedaimon, somehow (it seems counter-intuitive, at least from a regional diplomacy point-of-view, to start it off as violently as the version in which Klytaimnestra is first widowed in order to be [re]married), I don't see that Agamemnon has anything to gain by marrying Klytaimnestra.
Based on all this, his status does not seem to depend on his wife, royal or otherwise. Granted, of course, that upon feeling slighted enough by him, this wife does organise for him to be slaughtered when he returns home from Troy, so in that sense, well... his status depends on her a frightfully good deal.
Klytaimnestra's Dowry
At the end of the 400s BC Euripides wrote a play entitled Iphigeneia at Aulis. A fairly major character thereof is a quirky unnamed individual known simply as the "Old Man."
In the play's introduction scene, a discussion comes up between this character and the Argive king Agamemnon, in which we find out that the Old Man is a slave belonging to the royal household of Argolis. Via a bit of expository dialogue in Lines 42-45—addressed here to Agamemnon but really directed at the audience—the slave reveals to us that:
Your old father-in-law Tyndareos [Tyndareus] gave me to your wife Klytaimnestra as part of her dowry, to be her loyal servant.
The Old Man makes a number of subsequent appearances on stage. In his final scene he reminds us—this time addressing Klytaimnestra—that, "as you know, I was part of your dowry when you married King Agamemnon" (Line 869).
These are the only explicit ancient references I have found to the content of Klytaimnestra's dowry.2 The elderly slave makes no mention of what (or who) else accompanied him at the time of the transfer in question.
Helen's Dowry
It is arguably a lot messier to determine what Helen's dowry was. It could perhaps be said that her dowry was the entire kingdom of Lakedaimon, into which Helen had been born a princess, but I don't see any explicit mention which draws that conclusion.
It is a (maybe surprisingly) common trope in Greek mythology for a foreign man to marry into the royal family and thus become king, succeeding his father-in-law on the throne. Typically this is because the father-in-law has produced no direct male heirs or he has survived all of his male issue.
(See for example Perseus inheriting the throne of Ethiopia, which he never takes but transfers to his firstborn son Perses; Pelops—grandfather of Agamemnon and Menelaus—becoming king of Pisa after Oinomaos [Oenomaus]3; Alkathous [Alcathous], a son of Pelops [and thus uncle of Agamemnon and Menelaus] becoming king of Megara after Megareus; Sikyon [Sicyon], another son of Pelops, succeeding Lamedon as king of Aigialia [Aegialia], which he renamed Sikyon after himself; and [Priam's ancestor] Dardanos [Dardanus] becoming king of Teukria [Teucria] after Teukros [Teucer].)
Are we to automatically understand from this that the kingdom which the outsider inherits is ordinarily part of his princess bride's dowry? If the answer to that is no, then we do not seem to be told—between the entire kingdom of Lakedaimon on the one hand, and merely a fraction of its resources on the other—where Helen's dowry begins and ends.
The Status of Menelaus
But let's assume that the answer to that is yes, and that the dowry in question includes the lion's share of Tyndareos' family wealth. How Helen and her own position should affect the status of Menelaus is still a murky thing to ascertain, not least because of the unprecedented saga in which the two of them get embroiled.
It obviously is an unusual nuptial negotiation setup to have all of Greece's most powerful and most important kings and princes suing for the hand of one princess at the same time with as much determination as the suits for Helen ended up garnering.
Nonetheless, even with all of that baggage to deal with, there is a scene in Iphigeneia at Aulis which suggests that Menelaus bears no obligation to recover his wife from Paris. At his first appearance in the play, Menelaus engages Agamemnon in a quarrel about the planned sacrifice of Agamemnon's daughter Iphigeneia to Artemis in order for the goddess to release the winds for the assembled Greek army fleet to sail to Troy and attack it.
Agamemnon has been trying to find a way of avoiding killing his daughter, and Menelaus berates him for it. They are interrupted by a messenger's announcement that Iphigeneia, who thinks that she's been brought to wed the Myrmidon prince Akhilleus [Achilles], has arrived.
Upon Agamemnon's genuine distress over Iphigeneia's arrival, Menelaus changes his mind and even urges Agamemnon not to go through with it. They could just disband the gathered army and forget the whole expedition. As he says:
After all, what is it exactly that I want? Is it marriage? Surely I could find another wife elsewhere! Gods forbid that I should choose to lose a brother to win a Helen! Should I exchange the good for bad?
He doesn't seem much concerned about losing the throne, if such a threat even exists. Another wrinkle to the question of Lakedaimon's throne has already appeared. Menelaus and Helen have a daughter, named Hermione, whom Helen left behind in Lakedaimon when she took off with Paris.
Is Hermione a secure heir at this point, whether her mother is a present and dutiful wife or if her mother has been taken by another man? Is it enough that Menelaus has had a legitimate daughter by the previous heiress to the kingdom, whatever it is that may have happened to said heiress?
Going by what seems to be virtually customary all over Greece and the Mediterranean Basin by this point, as far as the Greek myths go, all we would need to do is find an eligible (typically foreign and royal) bachelor for Hermione to wed and it's all good: he would be next king. And that need not mean Menelaus' displacement before his time is done.
As fate would have it, later on, after the saga has cooled down and Helen has been brought back, a stranger (in a sense) comes to town and marries Hermione. The groom is none other than Prince Orestes of Argolis, the son of Agamemnon and Klytaimnestra, and thus Hermione's own cousin, who will thus rule over both Argolis and Lakedaimon, in a consolidated position more powerful than either of the immediately preceding rulers of these kingdoms.
Going back to Menelaus: perhaps the lines are blurry enough between wounded pride; the famous Oath of Tyndareos; and doubt as to whether his position on the throne is secure after he has been so vilely cuckolded and robbed of "the treasures" (which are usually invoked as part of what the Greeks demand to be returned along with Helen) by an honoured guest in his house, that his brother Agamemnon is convinced he must after all go through with the sacrifice of Iphigeneia. Thereafter, of course, come the long years of war far from home.
The Status of Agamemnon
While Menelaus' status might be quite open to interpretation, I think that Agamemnon's is much less so. Apart from the apparent politically diplomatic (and thereby strategic?) connection to his wife's family and its fame, Agamemnon does not appear to have anything to gain by marrying Klytaimnestra.
Agamemnon is primarily the king of the city of Mykenai [Mycenae], which position makes him the ruler of all of Argolis, a large northeastern swath of Peloponnesos [Peloponnesus] which contains the great cities of Argos, Tiryns and Troizenos [Troezen]. The rule of the city of Argos is divided among at least four kings (including Kyanippos; Sthenelos son of Iphis; and Tlepolemos son of Herakles) who reign under Agamemnon. By the time the Trojan War has ended (or shortly after) so have the reigns of these men, unceremoniously, while the throne of Agamemnon is still very much intact, to later devolve upon his son Orestes, who becomes even more powerful than his father.
Shortly after becoming king, Agamemnon led an army north of Argos, against King Hippolytos [Hippolytus] of Sikyon, a great-grandson of Herakles, and made him subject to Mykenai. According to Strabo (Geographia 8):
When everything fell to the sons of Atreus, Agamemnon, being the elder, assumed the supreme power, and by a combination of good fortune and valour acquired much of the country in addition to the possessions he already had; and indeed he also added Lakonia to the territory of Mykenai. Now Menelaus came into possession of Lakonia, but Agamemnon received Mykenai and the regions as far as Korinthos [Corinth] and Sikyon and the country which at that time was called the land of the Ionians and Aegialians but later the land of the Achaeans.
Lakedaimon was part of Lakonia. Strabo's account here essentially nullifies altogether the idea that Menelaus was dependent on his father-in-law for his position as king of Lakedaimon. Apollodorus' Epitome 2.15, however, tells a somewhat different story in which Tyndareos, at least for a time, looked after Agamemnon and Menelaus when they were in exile from Argolis, saying that he brought them back from their wanderings in Sikyon and Aitolia [Aetolia] so that they could reclaim their birthright. Perhaps there was some symbiosis taking place in their relationship with Tyndareos.
Beyond all that, Agamemnon held territory near Pylos in Messenia (on the direct opposite side of Peloponnesos from his home base), at least seven cities' worth, and, according to Strabo, he imposed his rule over all of Peloponnesos, like his grandfather Pelops, who had (re)named the vast southern peninsula of Greece, "Pelops' Island," after himself, its emperor.
Not every version of the story of Agamemnon's wedding to Klytaimnestra has her previously married so that her husband and child are killed by Agamemnon before he takes her for himself. In this case he does not necessarily force her to marry him, at least not any more so than women would have had very much of a say in who they married, or when.
Either way it is remarkable that Agamemnon, in the violent courtship version, manages to get away with the murder of Klytaimnestra's former husband, and the way he gets away with it is noteworthy. In Iphigeneia at Aulis 1150-1155, Klytaimnestra's divine brothers the twin Dioskouroi [Dioscuri], who had departed from the world of mortals years before, are so enraged at Agamemnon's slaying of their brother-in-law and their nephew that they actually leave Zeus' side in heaven or on Mt Olympos [Olympus] and come charging on immortal horseback against Agamemnon.
Inexplicably, Agamemnon pleads with Tyndareos to help him out, and Tyndareos obliges him, not only convincing the Dioskouroi to let the offence go but also marrying Klytaimnestra off to her husband's killer! Either Agamemnon had something on Tyndareos or they must have had a stellar relationship.
None of these events, however, seem to add to Agamemnon's clout beyond the ways already aforementioned. He was already a rich and powerful king and his dominions continued to increase after he got married, evidently in spite of the marriage. As already pointed out, his brother already ruled the southeastern portion of Peloponnesos (which Strabo claims Agamemnon had given to him in the first place anyway), which certainly would only have served him further.
Again, not that any of this saved him from a bedchamber assassination plot when he got back home from the war in Asia.
Notes
1. Okay, so technically the true ancient spelling of this character's name, such as it occurs, for instance in Iphigeneia at Aulis, actually has no N in it. It is really Κλυταιμήστρα = Klytaimestra/ Clyt[a]emestra. The more common spelling in modern times (with the N), such as I have opted to use in this Answer, supposedly does not occur before the 700s AD, at the earliest.
2. It would be interesting if there are after all other references which provide further detail.
3. In the case of Pelops and Oinomaos, the only way for Pelops to marry Oinomaos' daughter is by, first, defeating his father-in-law-to-be in a chariot-race and then killing Oinomaos, which tasks Pelops succeeds in performing (even if he cheats a little bit in order to win).