You are not your personality. Your personality is the collection of opinions, likes and dislikes, values, beliefs, etc etc. It constantly changes moment to moment, and only exists as a result of previous thoughts and biological processes. That dies when your brain dies. That gets damaged when the brain gets damaged. YOU, as you know yourself ceases to exist when you die.
Like TheColonel said, there is a more subtle level of consciousness that pervades every moment of your life. It is the "watcher," or the simple awareness of existence. Sit and watch your thoughts as they flit through your head, whatever is watching that gong show without getting disturbed is your watcher. It doesn't change and it exists really whether you like it or not. Meditate a little and you'll see what I mean. Even if you're drunk, high, or whatever, its still there, although you may be more inclined to notice shiny stuff. It's there even when you sleep, lucid dreamers can attest to this, as will really advanced meditators. Although there's no way to prove it, I would bet that people with brain damage or memory problems also live with this pure awareness.
That part of you, I think, can survive death, or lasts past death.
On to why I believe in the afterlife:
There are cases of people who remember past lives. I agree many are just cranks, or pretending to be important, or they are pretending in order to make money somehow. This being the case, I still believe that some may be actually telling the truth. Several Buddhist monks throughout history claim knowledge of past lives. Reincarnation seems to me the most accurate notion of the afterlife, given what I've seen so far. I don't believe in the classical Christian interpretation of the afterlife, simply because it does not make sense to me that one lifetime is worth an eternity of punishment or happy god-watching. But thats a different argument
