Can I do an installation from the OS disc and then start up from a time machine back up?
Best bet would be to erase the drive and reinstall OS X from scratch, then restore your data from Time Machine. This will all be a waste of time though if the TM backup also contains Vuze. You'd have to pick a restore point before Vuze or the Adobe download was installed, if that point is still on the TM drive.
If I do that I guess I will lose all my latest files etc?
Not if you back them up first. By that, I mean a manual backup. Don't use a flash drive as whatever is on the Mac may transfer data to it, which could then be pulled back to a fresh install. Instead, copy any documents you've created, your email, iTunes library, etc. into a folder on the desktop. Burn a CD or DVD.
Others will have to tell you what to backup for iTunes or Mail since I use neither.
Or, how to restore just those items from your current TM drive after reinstalling OS X from scratch so you don't pull Vuze back and whatever other malware was installed with that Adobe download. Which would eliminate the need for the CD/DVD. Though personally, I would create that disk just to be sure I have those latest files somewhere safe.