The modern-day Animus subject played by the player in Assassin's Creed II is Desmond Miles.
Desmond is a descendant of several historical Assassins, including Altaïr Ibn-La'Ahad and Ezio Auditore. The Animus allows him to relive genetic memories inherited through his family line, which provides the framing device for the historical adventures in the game.
In Assassin's Creed II, Desmond escapes Abstergo with Lucy Stillman and joins a group of modern Assassins. He uses the Animus to experience Ezio's memories, while the modern-day team searches for information useful in the ongoing conflict against the Templars.
William Miles is Desmond's father, not the playable subject in this installment. Warren Vidic is an Abstergo scientist, and Clay Kaczmarek is another Animus subject associated with later revelations. Those names belong to the same modern-day storyline but do not answer the question.