Will the Radeon 7750 work with this set-up?

Just about any dedicated video card will work with your system. Whether it makes sense in using a high end GPU with your proposed system is another topic.

What is your budget for your system?
Did you have any other special requirements for your system?
I would recommend a z68 based mobo, if you want to have the most flexibility with your system in the future, otherwise you can get by with a bit cheaper option, if needed.

No, I'm trying to find a dedicated gpu for my existing system, the specs I gave are what I'm using right now, heh.

* Well having the PSU make and model will help a lot too.
* Your mobo isn't as critical to know, but definitely helps to tailor other parts. You can either open up your case and see what the model # is that is screen printed (usually) onto the mobo or you can find the box that it came in and get the info off of it.
* I'll still need to know what your budget is for the GPU, so I can recommend the best for your $.

Well., this is what I found for my model: http://h10010.www1.hp.com/wwpc/ca/en/ho/WF06b/12454-124...

I could open it up and check the mobo, but how to I check PSU?

Here is your product specifications that I found on HP's website. It looks like a 250w PSU, so I'd recommend the 7750 (since it doesn't require any PCI-e power connectors to operate properly).

So to sum it up, you should be fine with the 7750 you list or any 7750 or a lesser GPU that doesn't require a 6 pin PCI-e power connector. Otherwise you would need to upgrade the PSU and buy a GPU too.