Can I use a passphrase as a private key? Is there a way to use a long (and possibly hashed) passphrase as a private key, so that all I need to recover my Bitcoins would be that password?

I was thinking of buying Bitcoins using a private key that's generated from a passphrase, and then deleting the Bitcoins software entirely. When I want to use my Bitcoins in the future, I will re-create the key using the password. This will eliminate the possibility of stealing my Bitcoins by hacking to my computer.
The simplest and most common way of doing this is to do SHA256(passphrase) and to use the result as your private key. You can e.g. use the "Brain wallet" tab on www.bitaddress.org which is all javascript so that you can download it and do everything offline for increased security.

Be aware though that your pass phrase must be really long and truly unique since if someone would use the same pass phrase they would have access to your money. The comment from Pieter Wuille above is worth repeating:


  Be aware that when you do this, you expose your wallet to the entire
  world to do a brute-force attack on your password. If many people
  start doing this, it may become very profitable to try to find their
  passwords. Software like Armory and Electrum have the ability to
  generate a determinsitic wallet from a 'seed', but this seed is still
  randomly generated, in order to be sure that it is hard to attack