You can't just go up and evict him if he hasn't done anything wrong. You can talk to him and see if he wants to move and voice your concerns about him living there, but you cannot get right of him. When his lease is up, you can decide not to renew it, however. I'd inform him of your decision right away and say you won't charge any early lease termination penalties if he decides to leave earlier if you want him out that badly.
But to answer your question, no, I do not believe you can evict someone for suspected illegal activity in another apartment. He hasn't done anything wrong.