Will this GPU work with my computer?

Yes it will work with your pc but tbh youmight need something along the 6770/6850 range for that cpu to alleviate a bottleneck.
If your set on the 6870 it is your cash and will definitely work on all accounts with your pc, power supply and monitor. Good luck and have fun

It will work, But you would also need to get a Power supply, the 1 that HP supplies in these pre built system are under powered for cards like the 6870, If you get a new Power supply, you should be all set, just don't got all cheap on a one tho, Make sure its got the Watts and the Amps that the 6870 needs to power up, also a good brand. Go cheap on a no name PSU and you could lose the entire system, its not if it fails, its when it fails.

As stated in the topic post, I have a Seasonic M12II 620 Watt, if that's good enough.

as OP stated he has the seasonic 620w, which wil be perfectly fine for his intentions. Just need to make sure you have 2-6 pin connectors for the new card.

I have 1 6 pin and another that is a 6/8 pin. That would work right?

That should work perfectly.

yes it will, the 6+2 pins just tie the 2 pin back and plug in the 6, easy.
Good luck with your upgrade