I think it would be depressing to be the last person on Earth. You'd have no one to talk to or share what you're doing with except yourself.
With that being said..artificial insemination would be an interesting start, but wouldn't you need a host to actually breed? I guess I would have to conduct a lot of research on that.
But for fun, I would go into all the really expensive stores and try on all the clothes and shoes and choose an awesome outfit to wear...which would be sort of pointless since I'm the only person left, I could walk around naked I suppose.
But it would be fun to shop without worrying about the prices. Then I would start heading south to better weather. I wouldn't want to get stuck in a Chicago winter without heat.
I would live on the beach until I went completely insane with loneliness and comitted suicide or died from lack of food or too much smoking. If I'm the last person on earth I would never have to hear about second-hand smoke again.
I guess I like to share my accomplishments and things I do with other people. If someone isn't around to talk with and learn from, what's the point? I wouldn't be a good candidate for the 'last person on earth' position. The human race would definitely go extinct.