Jeez... Even thinking about becoming an immortal on a hypothetical level unnerves me to no end.
I am pretty sure that I would have some sort of mental breakdown at first, knowing that I am going to outlive everyone I ever love and there will be a point where it gets very dangerous for me to be out in society.
I know that I'd want to try to do something worthwhile with my time, like learn new things, become more skilled in battle, and so on... So I can, ideally, do something worthwhile for the world. But even that would be hard considering that if any government hears about there being an immortal being walking around, you can damned well bet they'd dissect you for scientific purposes and/or put you to work as an assassin.
However, the worst thing about all this is that I would have to likely fake my own death right off the bat. I would have to abandon my family since, despite how much I love them, this is something I could never tell them about.
Past a certain point, though, I would try to find a way to kill myself. I don't know when, but probably when my perception of time goes completely to crap (since it does speed up as you age/live on) or it becomes too difficult to move around and exist as normally as possible. Like another poster said, I'd have to make certain that every ounce of me is destroyed on a celular level? So I'd probably go pitch myself into some lava and pray that I do not come back from THAT.
Also, Cracked did a nice article about this topic.
http://www.cracked.com/article_18708_5-reasons-immortality-would-be-worse-than-death.html
