Computer capable of running one of these GPUs? Specs Detailed below.

You are mistaken - room inside is huge problem, because you are using BTX design, not every card can fit inside.

Let me show what I mean.


THIS POST APPLICABLE TO MT (MINI TOWER) FORM FACTOR ONLY.

Unfortunately no card that is more than 7.5 inches long or 2 slots wide will PHYSICALLY FIT inside the case - it was found that only single bracket video card can fit due to BTX MOBO, where card internals facing upward towards PC internals, not ATX, where dual card design is blocking another PCI slot by facing downward.
If you prefer the card with beefier fan, then the length of the card must not exceed 7.5 inches, otherwise only "skinny" single bracket, single slot cards can physically fit.

Pictures below will prove my point.

View shows no room for dual bracket



Single bracket only, or you have to cut dual bracket in half



This is the other way to show my point



Here you can clearly see only single place for the bracket



Visual explanation



7.5 inches explained, also shown "obstacles"



This is the picture of actual "skinny" single slot video card (not sure which model)- perfect, "glove" fit.



So, I found 2 GDDR5 7750 cards and one DDR3 4GB (GDDR5 is better than GDDR3, even if GDD3 offers 2 GB RAM or more), which confirmed in reviews to fit Optiplex gx620, 745 and 755 cases (the difference between those models are different processors).

Skinny, flimsier in terms of construction, completely single slot XFX 7750 http://www.amazon.com/XFX-DisplayPort-PCI-Express-FX775AZNP4-FX-775A-ZNP4/dp/B007Z3T5JC

A bit beefier cooler, not as skinny, more durable in construction PowerColor AX7750 http://www.amazon.com/PowerColor-AX7750-1GBD5-DH-Video-Graphics/dp/B00775OC28/ref=sr_1_1?ie=UTF8&qid=1362783129&sr=8-1&keywords=AX7750+1GBD5-DH

If you need Video card for working with high resolutions, BUT NOT FOR GAMING, PowerColor AX7750 4GBK3-H Radeon HD 7750 4GB is DDR3 card with huge 4GB memory, but is is not as fast as DDR5 and loosing in games http://www.newegg.com/Product/Product.aspx?Item=N82E16814131486&Tpk=7750&IsVirtualParent=1
Here is review of this card http://news.softpedia.com/news/PowerColor-Launches-Single-Slot-Radeon-HD-7750-4-GB-Video-Card-292712.shtml so this one is not for gamers!


Next will follow maximum possible upgrade, 7750 is weaker...

So far I found only 2 7770 video cards which will fit inside - PowerColor AX7770 http://www.newegg.com/Product/Product.aspx?Item=N82E16814131477

HIS H777FN1G Radeon HD 7770 very tight fit, almost touching heatsink http://www.newegg.com/Product/Product.aspx?Item=N82E16814161433


7750 cards do not require power supply upgrade, but 7770 are.

However first of all I would like to add comment on PSUs, please check this list for choosing PSU to prevent doorstep creation or even worse - fire in your house!

Also, I would recommend getting modular PSU to avoid cable cluster in small cramped boxhttp://www.newegg.ca/Product/Product.aspx?Item=N82E16817139050&Tpk=CORSAIR%20CX500M%20500W%20ATX12V%20v2.3%20SLI%20Ready%20CrossFir&IsVirtualParent=1

When choosing PSU follow this guide http://www.eggxpert.com/forums/thread/323050.aspx


There are so many options available here so I would just recommend one of the good one CORSAIR CX500M (modular), which has just enough power http://www.newegg.com/Product/Product.aspx?Item=N82E16817139050 Also, with modular you can get rid of extra cables, believe me there are a lot of them.
You need 500 watt at least, stronger is better depending on your money.

Anything else I can help you with?