You don't need a new computer, but a new PSU might be in order, also, 700w is maybe a bit high, while the one you linked is inexpensive, PSU is the one place you don't want to skimp in quality, a good one can keep your computer running like a champ for years with no problems, I'd recommend a 550w Seasonic or XFX. You should be able to get one for around 50$ and that'lll be enough for that card and the rest of your system. I've got a 650w XFX that I bought about 5 years ago, give or take, and I'm litecoin mining a 7870 XT and a 7970 XFX Double D. So 650 is more than enough for a 760. If you have some patience you can wait for rebates, I bought my parents a 650w modular XFX pro edition for like 39$ after the rebate. That was I think last month before the black Friday sales.
Here's a tiered list of PSUs, mind you, it's not ALL manufacturers and models, but it'll give you an idea where you're sitting as far as quality goes.
http://www.eggxpert.com/forums/thread/323050.aspx