Will the 5770 run on this computer?

Instead of use 2 PSU on the same system, u better buy a new PSU with better quality...
There's no other way, use 2 PSU or get a new one...  

Well, I'm not the most tech savy person around, the most I've done is install a few GFX cards. I'm quite sure I would probably completely destroy any computer I tried to change the PSU on, and my budget is already stretched.

It's okay, u learned from what u did, right?  
If u resist to use 5770 on that PSU, u will regret it later if something bad happened. Believe me, PSU is the most important part of your system...

I might replace it, but two of them would be my first choice. Do you know if that would cause any issues? Also, do you think this card is overkill for this computer?

I can almost guarantee that the 5770 will not run on that PSU, HP stocked it with a 4350 which is incredibly low power, if they could have gotten away with using a 250W unit they would have, adding in a 5770 which is significantly more powerful would almost certainly overtax the unit.

Replacing a PSU isnt all that difficult, you just unplug all of the cables, undo 4 screws, remove the unit, put in new unit, put in 4 screws, plug in all of the cables, done. Getting it out of the case is the tricky part on prebuilts with cramped towers but it is still possible, i did it a couple years ago on my HP, you may need someone who is good at puzzles to help you get it out though, hopefully their towers are more spacious now than they were back in 02.

no, it's not overkill, it's a good card...
The issue is your case is looked bad..   with 2 PSU on it...

Alright, any recommendations for a PSU? I'm going to be using this computer for a few years.

budget?

As low as possible, lest start at $50. I already broke my original 600 dollar budget.

If u want quality:
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
i know it's 80$

it's hard for me to choose...
Anyone could help us?

I think that's over my budget, that's why I just wanted to use 2 PSUs. Would that work? I don't care how it looks.

http://www.newegg.com/Product/Product.aspx?Item=N82E168...
That is closer to your budget and would power just about any single GPU card without problem. The modular nature will definitely help out in a cramped OEM case too.

I don't know, I probably can't afford the 5770 if I get a PSU.

I did the math, all I need is a 400W PSU too run this computer with the 5770. So now I'm looking for a good 400W PSU.

True but it is always best to have about 50 extra watts on the psu to ensure you have enough. Going with the exact amount can almost always be a bad choice.

corsair 450 watts

http://www.newegg.com/Product/Product.aspx?Item=N82E168...
Can run new games medium-low quality, and older (2006 - 2008) games med - high quality, and is cheaper than having to get a new power supply.


OR

http://www.newegg.com/Product/Product.aspx?Item=N82E168...
and
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

You probably wouldn't even need a 5770 either way, and a 5750 would suffice.