Here's yet one more recommendation for CarbonCopyCloner (or SuperDuper) instead of Time Machine.
A cloned backup will give you one advantage a TM backup simply -cannot- give you -- a fully bootable backup drive. By "fully bootable", I mean that you can connect it, boot to it, and have everything available to you, ready to go to work -- all your applications, all your data.
Immediately, and without a lengthly "restoration process".
(Aside: of course, you will still need to diagnose the problem with your internal drive and repair or replace it, but you can still have a fully functional Mac while such operations are underway).