moonshines
Lag of a brother? (1-6)

Then he asks why he (or anyone in his situation) is labeled with such a low ("base") title as "bastard," when he is just as physically fit and mentally blessed as any young man born to a mother.

Why bastard? wherefore base?
When my dimensions are as well compact,
My mind as generous, and my shape as true,
As honest madam's issue? (6-9)

He wonders that such a distinction can be made when so many children who are born "properly" are absolute fools, being "made" the same way as those who are "illegitimate."

Who, in the lusty stealth of nature, take
More composition and fierce quality
Than doth, within a dull, stale, tired bed,
Go to the creating a whole tribe of fops...  (11-14)

At this point, Edmund directly addresses the absent Edgar, his "legitimate" brother (and he chuckles at the use of the word), saying that he needs Edgar's land (and all that goes with it). He also states that he is loved by their father just as much as Edgar is. If Edmund's "plot" is successful, he—the illegitimate son—will "top" the "legitimate" son. And he calls on the gods to be supportive of bastards.

Legitimate Edgar, I must have your land:
Our father's love is to the bastard Edmund
As to the legitimate: fine word,–legitimate!
Well, my legitimate, if this letter speed,
And my invention thrive, Edmund the base
Shall top the legitimate. I grow; I prosper:
Now, gods, stand up for bastards! (16-22)

It is important to note that Edmund is something of a villain. He will go to great lengths to destroy any commerce (good relations) among his family, and within King Lear's family. At this point, he will do so deceitfully in an attempt to take Edgar's lands (and title); he is going to forge a letter that will make Edgar look like a traitor, planning to kill his father, the Earl of Gloucester. Edgar has no such plan, but Edmund will be "base" in the way he acts, if not in the station to which he was born.

• In King Lear, Edmund says he is a victim of culture, not nature: he is not born evil, but made evil by society shunning him and all bastards.
• He is the foil for Edgar.
• Edmund is a Bastard, an illegitimate son, dispossessed.  He burns with resentment. He can’t have what he wants, so he lashes out to hurt those around him. His deeds are often for effect – he wants to provoke action in others. He proudly announces his rebellious dealings.
• He is the "bad brother," like the biblical Cain.
• He is a villain, joining Regan and Goneril, Oswald, and Cornwall as the antagonists in the play.
• He is an alazon, an impostor who thinks he is better than he really is and deserves more than he really does.
• His appearance is young, dark, and lusty.
• His characteristics are ambition, hunger for power, self-importance, manipulation
• He is motivated by cruelty, deceit, and power
• He achieves his aims by leading Gloucester to believe that Edgar, the earl’s legitimate son, is plotting to murder his father.

Since accessteacher used Act III, scene 3 and you did not specify which soliloquy, I shall talk about his first one which begins Act I, scene 2.

In th beginning of the play, Glouchester and Kent are chatting and his father makes rude jokes about his inception.  Edmond is there but says nothing.

It is at the beginning of the second scene that Edmond tells us how he really feels.

I had the privilege of being in a workshop conducted by Cicely Berry, the head of voice for the Royal Shakespeare Company where we worked on this speech. 

First, she asked us to free associate the word nature and we responded with beauty in nature.  She then asked us about nature in the negative and we responded with things like severe storms, floods, earthquakes, etc.  She then asked us which nature Edmond was calling his goddess.

She then asked us to think of something in modern society which would make a person an outcast.  The best we could come up with was AIDS.  As she explained it to us, being a bastard in that world meant being an outcast.  A good example of this is Glouchester's attiude toward him especially compared to Edgar.

Being illegitmate brands him and to demonstrate this, she choose one of the young men in the group to repeat over and over until she told him to stop, the lines about being branded a bastard.  She then approached two other young men and a young woman to whom she gave her pen then whispered her instructions to them.

These three people approached the other person and the two men grabbed him and wrestled him to the floor and the girl straddled his chest and using the reverse side of the pen, began to "brand" his forehead with the word bastard.  Throughout it all he continued to repeat the given lines but once they wrestled him to the floor, his voice became angry.  She stopped the excerise and asked him to say the lines and he was extremely convincing.

Edmond is an angry young man but he knows that to "win" he must outsmart his father and brother.

You have not specified the act and scene in which the soliloquy appears, so I am assuming you are refering to his soliloquy in Act III scene 3, where Edmund sees his opportunity to betray his father openly, having already betrayed his brother in secret, thanks to some information that Gloucester shares with Edmund regarding the invasion of a French army to revenge the "injuries the King now bears." Gloucester's own removal to attend to Lear and his instructions to Edmund give him the perfect opportunity to make his move and reveal this information to Cornwall, anticipating his inheritance of his father's title, wealth and estate once he is put to death:

This courtesy forbid thee shall the Duke
Instantly know, and of that letter too.
This seems a fair deserving, and must draw me
That which my father loses--no less than all.
The younger rises when the old doth fall.

Note how Edmund is presented in this soliloquy as a backstabbing, selfish and opportunistic individual who will stand at nothing to succeed in his goals and aims. His perceived slighting at being the bastard son of Gloucester causes him to plot his own father's downfall, and we can imagine his gloating as he delivers the final line of this soliloquy.

Edmund is a villain in King Lear first because he is deceptive. He convinces his brother and his father that he has their best interests at heart when, in fact, he is working to harm both of them. He is so filled with seething anger and resentment because of his status as an illegitimate son that he wants to destroy Edgar, the legitimate heir to the earldom of Gloucester.

Edmund doesn't simply seethe but acts, showing his evil and devious nature though a duplicitous plot in which he plants a fake letter for his father to find showing Edgar's "treachery." Then, pretending to be Edgar's friend, he warns him to flee and hide from his father's wrath. Neither figure has any malice against him, and his father accepts him openly as his son; but Edmund is nevertheless filled with hate. He goes so far as to wound himself, saying he was attacked by Edgar, to manipulate his father into turning on Edgar.

Edmund also plays Goneril and Regan off against one another, as both are in love with him. He cold-bloodedly states that one must die, saying "Neither can be enjoyed / If both remain alive." He also cold-bloodedly arranges to have Cordelia hanged in her prison cell at the end of the play, though he later repents of this and is too late in trying to stop it.

Edmund's destructive and calculated actions show that he is evil, and this evil springs from a mindset that can't accept that life is unfair. He is a "cup half empty" person: he can never see what he has, such as the status of an earl's openly acknowledged illegitimate son, a privileged if imperfect position. Like Goneril and Regan, he is without gratitude. Many of the poor that Lear finally comes to have compassion for would jump at what Edmund has, but all he can see is what he doesn't have.