It may not be possible to replace the power supply in your HP computer, except by buying one through HP, and they may not even offer that for this model.

There are two reasons:
1. Some OEM PCs have propriety motherboard connectors which require you use a propriety power supply to connect to; they aren't physically compatible with off-the-shelf PSUs.
2. Because this appears to be a small form factor computer, and particularly because it has a 250w powersupply in it, the space inside the case may simply not be big enough to fit a regular off-the-shelf PSU.

The only way you're really going to answer these questions is by either opening the case, or trying to find someone on the internet who has the same computer and can open the case and tell you, or your ring up HP support and somehow get someone technical that can answer your questions (and since you yourself aren't technical, this may not help).

The Radeon 7750 is a single-slot GPU which draws all power from the PCIe slot. It's certainly not "really shitty and not worth anything".

Obviously, however, Tempz is being retarded and there is no way he's getting a new card without a PSU upgrade. I believe that HP has a standard ATX power supply, but check the manual to make sure. Also note that Skyrim is a CPU-bound title, so your i3 will not get you far.

I'm willing to install a new cpu/battery... although i'd rather get someone else at a independent computer store to do it.

You need a new video card as well.
That makes PSU, video card and CPU. That's almost as much as a whole new computer.
Clearly your computer wasn't built for gaming so I'd consider selling it and buying a gaming machine.