In your opinion, how do Duke Senior and Duke Frederick contribute to the meaning and dramatic effect of As You Like It?

Duke Frederick and Duke Senior embody one of two major conflicts between brothers in Shakespeare's As You Like It. The other conflict is between two younger men, Orlando and his older brother, Oliver. During the course of the play, these two conflicts intersect and provide a major impetus for much of what occurs in the play.
The dramatic importance of the conflict between Duke Frederick and Duke Senior is that it contributes directly to the major romantic plot line of the play between Duke Senior's daughter, Rosalind, and Orlando.
Before the opening of the play, Duke Frederick has deposed his older brother Duke Senior, usurped his power and his property, and banished him from his dukedom. As the play begins, Duke Senior and some of his loyal followers are living off the land in the Forest of Arden.
In the interim, Duke Frederick has become concerned about Rosalind's influence over his own daughter, Celia, and banishes Rosalind from his court.

DUKE FREDERICK: [to ROSALIND] Mistress, despatch you with your safest haste,And get you from our court.
ROSALIND: Me, uncle?
DUKE FREDERICK. You, cousin:Within these ten days if that thou be'st foundSo near our public court as twenty miles,Thou diest for it.
ROSALIND: I do beseech your grace,Let me the knowledge of my fault bear with me . . .
DUKE FREDERICK: . . . Let it suffice thee that I trust thee not. (1.3)

Celia is distraught at the thought of being separated from her dearest cousin and best friend and vows to go with Rosalind.

CELIA: . . . Shall we be sund'red? shall we part, sweet girl?No; let my father seek another heir.Therefore devise with me how we may fly,Whither to go, and what to bear with us:And do not seek to take your charge upon you,To bear your griefs yourself, and leave me out;For, by this heaven, now at our sorrows pale,Say what thou canst, I'll go along with thee.
ROSALIND: Why, whither shall we go?
CELIA: To seek my uncle in the Forest of Arden. (1.3)

Meanwhile, Orlando learns from an old family servant that his brother, Oliver, intends to kill him.

ORLANDO: Why, what's the matter?
ADAM: O unhappy youth,Come not within these doors; within this roofThe enemy of all your graces lives:Your brother . . . Hath heard your praises; and this night he meansTo burn the lodging where you use to lie,And you within it: if he fail of that,He will have other means to cut you off;I overheard him and his practices.This is no place; this house is but a butchery:Abhor it, fear it, do not enter it.
ORLANDO: Why, whither, Adam, wouldst thou have me go?
ADAM: No matter whither, so you come not here. (2.3)

Orlando and Adam flee to the Forest of Arden, of course. In due time, Orlando meets Rosalind in the forest—who he doesn't recognize because she's disguised as a young man (but that's another story)—and the romantic plot of As You Like It takes center stage.
The conflict between Duke Frederick and Duke Senior is established at the beginning of the play. Once the conflict serves its purpose of putting into motion the romantic plot with Orlando and Rosalind, it's barely mentioned again until the end of the play, when Duke Frederick, having undergone some kind of religious conversion, restores his brother, Duke Senior to his rightful place:

JAQUES DE BOIS: . . . Duke Frederick, hearing how that every dayMen of great worth resorted to this forest,Address'd a mighty power; which were on foot,In his own conduct, purposely to takeHis brother here, and put him to the sword:And to the skirts of this wild wood he came;Where, meeting with an old religious man,After some question with him, was convertedBoth from his enterprise and from the world;His crown bequeathing to his banish'd brother,And all their lands restored to them againThat were with him exil'd. This to be trueI do engage my life. (5.4)

Order is restored to the dukedom, and the play ends with four happy marriages, none of which would likely have happened without the conflict between Duke Frederick and Duke Senior.