How can I keep a process alive after closing the putty session?

^Z
bg %1
disown -h %1

The '-h' makes the process immune to SIGHUP when the session completes.

I use screen for that kind of stuff. Actually sometimes I just leave it on quite a while so I can get back to what I was doing.

Use the nohup command. Just prefix it to your command and it will daemonise them so that they won't stop when you log off/terminate your shell session. The standard output will by default be in a file called nohup.out. Check the manual page for nohup(1) for more information.