Soliloquy is used to keep the audience informed as to what particular characters are thinking and, in this case, plotting.
In all three scenes, Richard is shown to be 'mis'hapen' but very articulate. In scene 1 Richard tells the audience that he is more suited to war than peace because of his deformity and in times of 'romance' is 'determined to prove a villain'. So here the audience's introduction to Richard's cruel, almost abhorrent, personality is formed.
In scene 2 he carries this further by convincing Anne, whose husband and father in law he has had a hand in killing, to accept a ring from him. His soliloquy here reveals his gloating nature. He is very sure of himself and makes sure the audience hears about his plans and schemes as well as how clever he is in achieving them.
By scene 3, not only has has he proven himself to be articulate and scheming, but also murderous as he plots to have Clarence killed. These traits are taken to the ultimate degree throughout the rest of the play.
With Richard, soliloquies become an even more important tool than they are with most characters, because his personality is so deceptive and dissembling. He is not truthful or consistent with any of the other characters, and so to understand his true character and motives, it is important for the audience to see him in the privacy of his own mind.
The opening soliloquy also helps to build the complex relationship that Shakespeare intends for the audience to have with Richard. Not only are we made privy to his future plans (and the future course of the play), we also see him as an isolated individual from the society he is in. This helps the audience to develop a connection with Richard, which then allows us to sympathise with him, to admire him and to laugh with him later on in the play.
Without the soliloquies, the audience's relationship with Richard would not be half as complex as it is, nor would Richard be as intriguing as he is. Understanding Richard's character is crucial to understanding the plot, themes and other characters.