Looking for best graphics card I can use for cheap price compatible with my specs

You can go higher than that if your PSU supports it, what is the PSU? If it is 300watts it should handle the 7750 fine.

Your preconception of how graphics card series work is a little off. The 5000, 6000, 7000, and now Rx 200 series graphics card are just the new line of GPUs that come out each year. Within each line there are different tiers of GPUs. Within the 5000 series there are super high end GPUs and within the 7000 and Rx 200 series there are low end GPUs.

So for example, the Radeon HD 7750 is a low end card in the 7000 series. In the 6000 series, a high end card is the Radeon HD 6950. Guess which one is more powerful? It's the HD 6950. In the 5000 series a high end card is the HD 5950. It's more powerful than the HD 7750 as well. The first digits of the AMD GPU naming system merely specify the series. It doesn't specify performance individual card performance relative to any other series. Most of the time, a new series means that it'll perform better than its equivalent in the previous series. So for example, an HD 7750 will outperform an HD 6750, but it won't necessarily outperform an HD 6850, and it certainly won't outperform an HD 6950. You'll have to go a bit farther back (around the HD 4000 series and earlier) before low end cards of a recent series start to outperform higher end cards of an older series.

So there's little call to spend money on a previous series when you can just get a low end card of a newer series, unless of course you're in the middle of a transition between 2 series (which we are for AMD). Typically, aside from performance enhancements, cards of a newer series also have other benefits such as support for a newer version of DirectX or better power efficiency or better suited for newer drivers with certain games. So really it's always good just to buy within the latest series. Right now, the latest desktop series in the Radeon HD line (and also the last) is the 7000 series. But in fact, tomorrow (which is in T-minus 17 minutes where I am) the newest series comes out and it completely does away with the Radeon HD line. Instead tomorrow we'll begin to have the Radeon Rx 200 line. The naming scheme is different, but as it turns out, these cards are the same as the Radeon HD 7000 line just rebranded and tweaked a bit. They'll offer slightly better performance, but nothing too much in the way of new features I suppose. Though I do need to read up more on them.

So for now, it's safe for you to buy from the low end or mid-range spectrum of the Radeon HD 7000 series. The Radeon HD 7700 portion is entry level, the 7800 is mid-range, and the 7900 is high end. With a Phenom II X6 1045t CPU, which is an outdated but still good processor, you can definitely handle something mid range, i.e. 7800, or even something high end, i.e. 7900, assuming your power supply is efficient. A Radeon HD 7870 GHz Edition ($150 - $200) is a good buy for you, or you could try for the R9 270X ($200). You won't receive any bottlenecking with either with that CPU. However, you're on a budget, so I'll list some low end cards. The Radeon HD 7750 ($80 - $90) is a tad weak though, but you could totally go for an HD 7770 (~$100) or an HD 7790 (~$120) or an R7 260X ($140). Those are all low end, and the last 2 sort of, kind of blur the lines between low end and mid range. But if you can afford the latter R7 260X, then I recommend you just spend another $10 and go for a $150 Radeon HD 7870 GHz Edition. And again, you don't have to worry about bottlenecking.

Just remember for $90 you can not get any of those let alone a adding in a PSU to supply them with power.

@Deus the HD5xxx series had the HD5870 as the high end single GPU card and the HD5970 as the monster dual GPU card. AMD never released low end cards in the 7xxx series and continued to produce HD6450, 6570 and 6670 as entry level cards. The HD77xx was launched as a mainstream card http://www.anandtech.com/show/5541/

The classification is 4 parts Entry 6570/6670, mainstream 7750/70/90, mid range 7850/70 and high range 7950/70. And you are right about mid range.
HD7750 is about 50% better than the HD6670 so it is definitely a group up from that. http://www.techpowerup.com/reviews/ASUS/HD_7750/26.html

HD7750 is the best that runs on that 300watt HP PSU, you will need to upgrade the PSU for anything more powerful!

It's much simpler than you think. Don't let all those wires intimidate you. It's actually quite simple if you watch some YouTube tutorials and do a little research prior. It's how all of us here on Tom started out. I think most all of us here are self-taught. Anyway, as for a PSU, 300 W isn't enough, but 430 W is. Don't worry, you can always google a PSU Calculator to check, but here's a fantastic brand at a great price!
http://www.newegg.com/Product/Product.aspx?Item=N82E16817139026&nm_mc=KNC-GoogleAdwords&cm_mmc=KNC-GoogleAdwords-_-pla-_-Power+Supplies-_-N82E16817139026&gclid=CMyUtK6BmLoCFYai4AodkQcAiQ

And don't worry, that'll work for an HD 7870 GHz edition perfectly

UPDATE:
Just make sure that your PSU will fit your case. If you have the model number of your HP desktop we can tell you.

Make sure it is a quality unit and it is not very difficult if you pay attention when unplugging things.

Yep minitower std PSU. Solid 500watt budget unit from EVGA got good reviews http://pcpartpicker.com/part/evga-power-supply-100b10500kr

Yepp, OP you're good to go on ordering that thing along with your HD 7870 GHz Edition. Happy gaming!

500 watt budget EVGA unit http://pcpartpicker.com/part/evga-power-supply-100b10500kr review http://www.jonnyguru.com/modules.php?name=NDReviews&op=Story6&reid=351

7790 for less than $100 http://pcpartpicker.com/part/msi-video-card-r77901gd5oc

There is a big performance jump from the 7770 to even the HD7850.

Just for reference, here's a benchmark of Battlefield 4's Beta using most of the cards we discussed on the absolute highest settings for 1920x1080.



As you can see, your current card isn't even listed because frankly, it's pathetic (no offense). In fact, it probably doesn't even have the capability to run Battlefield 4 at all! Even if it does, there's no way it gets anything past the single digits. Meanwhile, the HD 7770 has a 4 min fps, 6 avg fps advantage over the HD 7750 and manages to keep its head in the 20s for the most part. That's not bad for the extra $10 - $20 over the HD 7750 if you ask me.

Then there's the HD 7870 (the superior, 7870 GHz Edition isn't listed here, but expect more frames than what's currently listed for the regular 7870). Compared to the 7770, it has a 12 min fps, and a whopping 21 average fps advantage over the HD 7770. An again, this is the weaker, non-GHz Edition, so you can expect an ever larger performance increase between those two. In my mind, it's worth the extra money, but you're right on one thing. The higher you go up the chart, the greater the price : performance ratio between cards becomes. The difference between a GTX 780 and a GTX Titan, for example, is merely 4/6 fps, just like the difference between the HD 7750 and the HD 7770. However, the price difference is a whopping $350 more! I'd still say it's worth the extra $70 for the HD 7870 GHz, because it means the difference between unplayable and playable frame rates. Of course, there's always the HD 7850 or the HD 7870 regular. But you have plenty of time to decide, there's no rush, and I'm sure that prices will continually drop.