Well, that is the thing - It won't work to a drive attached to windows. I am assuming that you have XP.
You are going to run into four problems why time machine may not work for what you describe.
1. The native file format on XP is NTFS. While a mac can read/write to an ntfs when it connects to a shared drive on windows it is messy.
Time Machine relies on the HFS file format to help know what files have changed and need to be backed up, as well as when to drop old files. This is why you drive will fill up pretty quick. If you format the drive to HFS, Windows can't read/write to it with out a third party app, and even then it is not a smart idea to use as anything more then for light use.
2. You have to set your mac to automount the windows drive. This isn't too bad with a mac network drive but sometimes it doesn't work well with a windows share. If the share isn't mounted, then there is no where for the backups to go.
3. You have to enter a terminal command to enable time machine to network shares. This feature looks like it was something that was shut off becouse it has problems. People have had very mixed results when using it.
4. Finally, and this is the big one. other then restoring files that accidentally got deleted, there is no way to get back to the remote time machine backup if you have a system fail. I am not talking about booting to it. I am saying if your drive crashes, and you reinstall the os and then pick restore from time machine backup. it won't work. Because you put it someplace that wasn't supported. for this reason alone i would think twice about this idea.
I realize that you are on a budget, and that dictates a lot of what you can do. On the other hand you may have to jump through a lot of hoops and ladders to get TM to work only to find out if there is a problem you have been lulled into a false security.