Why wont my computer start?

The battery is a good choice to start with. at least you have
eliminated the PSU, you are thinking on the right lines.

Failing this remove all except the CPU, RAM & Graphics and see if
you can get the bios to boot, you should get a message saying 'Please
insert Disc' or something to that effect... and you should be able to
get into bios as well.

In one post a similar fault was cleared after re fixing the CPU....
this is the last thing I would suggest to avoid any possible damage,
also when ever you remove a CPU always remove the old heatsink paste
or grey heat sink pad.... these pads are terrible good ol' Arctic
Silver paste (or similar) is a lot better, the CPU temp. is always a
few of degrees or so higher with pads and avoid the usual white stuff
you see in amplifiers and TVs etc this stuffs far from suitable for
the higher temperature and will soon dry out leading to problems.

Davy