I've ordered an external HD to use with Time Machine. After I make a Time Machine back up will I be able to access the files on a different computer or just on my computer?
You need to give us more info - you will be able to browse TimeMachine backups on a different machine - it will depend on what permissions have been set on the files as to whether you will be able to do anything with them (or not).
What i would suggest is that you try all your options and if it doesn't answer your questions just use the time machine and travel back in time and return your external HD