You wont need to back anything up, though I guess it doesnt hurt. Changing the PSU will not require an OS install. Just remember to earth yourself before touching the internals

PSU is pretty simple to change out. Switch everything off, remove power lead. Open the box up and disconnect all the leads trailing off from the PSU, keep a note of any that share leads, some fans etc run off a shared 4 pin lead. Once all the PSU leads are disconnected undo the four screws that hold it in, these are found on the rear of the case near the power lead point. Easier to do this with the case on its side so it doesnt drop down and hit the CPU fan, assuming its a full size ATX case with PSU at the top.

Drop in the new box and secure it in place, as its modular do the leads one at a time, starting with the main motherboard power supply, then CD/DVD, Hardrive and graphics cards, then follow up with the fans. If you can, get some plastic tiw wraps, when you have everything in place and made sure it runs, tie everything back and bundle things together. Really makes a big differenace both to the appearance and later component change, as well as improving cooling air flow.

Bite my shiny metal Tie Fighter!