Here's a great thing about buying a "Fix Kit" from iFixit -- you get all the tools you need (screwdriver, screwdriver bits, and spudger) you need to tackle the job. Don't attempt to do this with cheapo tools because you'll just cause mayhem.
This is why you need precision tools for the job!
This MacBook has seen a tough life, but despite multiple drops, copious cat hair, and evidence of tea ingress, it still works.
Start off by popping off the connector from the motherboard. The spudger makes lifting the connector off easy.
Here the mounting points on the battery have broken off, but I need to get the screws out to do the job. Here's the first screw...
... and here's the second mounting screw.
Pop in the battery, fit the connector...
...refitting one screw...
...and the other!
Pop the cover back on (this one was a bit banged up so needed some gentle persuasion to get it in place), do up the screws, and the job is done!