I used to play Spore using my friend's account because I borrowed it from him. Now that I've got a new computer, I bought my own copy of Spore. Is it possible for me to transfer the creations and the achievements I earned?

Creations, yes. Achievements, no.

How to I transfer the creations then?

You can transfer the creations you created by transferring My Spore Creations folder in Documents. You may need to drag them in the in-game sporepedia for it to work, but I think they should redownload when you next start spore. Note that if you should install the latest patch before doing that, and if your friend had, say, c&c, you need it to.