That is going to be long post, I tried to make it nice. Processor in BOLD type is most likely your model.
Inspiron 560 MT/560s tested by Dell processor list - Official Dell response.
Processor Number<>Processor Type<>cache<>Clock Speed<>Front Side Bus<>Package Type<>Thermal Design Power
Q8300 (560/560s)<>Intel® Core™2 Quad Processor<>4 MB<>2.50 GHz<>1333 MHz<>LGA775<>95W
Q8400 (560/560s)<>Intel® Core™2 Quad Processor<>4 MB<>2.66 GHz<>1333 MHz<>LGA775<>95W
Q9400 (560/560s)<>Intel® Core™2 Quad Processor<>6 MB<>2.66 GHz<>1033 MHz<>LGA775<>95W
E3200 (560/560s)<>Intel® Celeron® Processor <>1 MB<>2.40 GHz<>800 MHz <>LGA775<>65W
E3300 (560/560s)<>Intel® Celeron® Processor <>1 MB<>2.50 GHz<>800 MHz <>LGA775<>65W
E5300 (560/560s)<>Intel® Pentium® Processor <>2 MB<>2.60 GHz<>800 MHz <>LGA775<>65W
E5400 (560/560s)<>Intel® Pentium® Processor <>2 MB<>2.70 GHz<>800 MHz <>LGA775<>65W
E6300 (560/560s)<>Intel® Pentium® Processor <>2 MB<>2.8 GHz <>1066 MHz <>LGA775<>65W
E7500 (560/560s)<>Intel® Core™2 Duo Processor <>3 MB<>2.93GHz<>1066 MHz <>LGA775<>65W
E6500 (560/560s)<>Intel® Pentium® Processor <>2 MB<>2.93 GHz<>1066 MHz<>LGA775<>65W
E8400 (560/560s)<>ntel® Core™2 Duo Processor <>6 MB<>3.0 GHz <>1333 MHz<>LGA775<>65W
E7600 (560/560s)<>Intel® Core™2 Duo Processor <>3 MB<>3.06 GHz<>1066 MHz<>LGA775<>65W
E8500 (560/560s)<>Intel® Core™2 Duo Processor <>6 MB<>3.16 GHz<>1333 MHz<>LGA775<>65W
E8600 (560/560s)<>Intel® Core™2 Duo Processor <>6 MB<>3.33 GHz<>1333 MHz<>LGA775<>65W
450 (560/560s) <>Intel® Celeron® Processor <>512 KB<>2.20 GHz<>800 MHz<>LGA775<>35W
Q8200 (560s)<>Intel® Core™2 Quad Processor <>4 MB <>2.33 GHz<>1333 MHz<>LGA775<>95W
Q8400S (560s)<>Intel® Core™2 Quad Processor <>4 MB <>2.66 GHz<>1333 MHz<>LGA775<>65W
The following is a list of compatible processors per non Dell direct source (trusted source)
Conroe Kit, E4300, 1.80G, 2MB, 800FSB, L2
Conroe Kit, E4400, 2.00G, 2MB, 800FSB, L2
Conroe Kit, E6320, 1.86G, 4MB, 1066FSB, B2
Conroe Kit, E6420, 2.13G, 4MB, 1066FSB, B2
Conroe Kit, E6550, 2.33G, 4MB, 1333FSB
Conroe Kit, E6750, 2.66G, 4MB, 1333FSB, G0
Conroe Kit, E6850, 3.00G, 4MB, 1333FSB, G0
Conroe XE Kit, X6800, 2.93G, 4MB, 1066FSB, B2
Kentsfield Quad Core Kit, Q6600, 2.40G, 8MB, 1066FSB, G0
Kentsfield Quad Core Kit, Q6700, 2.66G, 8MB, 1066FSB, G0
Kentsfield Quad Core Kit, QX6700, 2.66G, 8MB, 1066FSB, B3
Kentsfield Quad Core Kit, QX6800, 2.93G, 8MB, 1066FSB, G0
Kentsfield Quad Core Kit, QX6850, 3.00G, 8MB, 1333FSB, G0
Yorkfield Quad Core Kit, QX9300, 2.5G, 6MB, 1333FSB, M1
Yorkfield Quad Core Kit, Q9450, 2.66G, 12MB, 1333FSB, C1
Yorkfield Quad Core Kit, Q9550, 2.83G, 12MB, 1333FSB, C1
Yorkfield Quad Core Kit, QX9650, 3.0G, 12MB, 1333FSB, C0
Yorkfield Quad Core kit, QX9650, 3.0G, 12MB, 1333FSB, C1
Wolfdale Dual Core kit, E7200, 2.53, 3MB, FD, 65W,M0
Wolfdale Dual Core Kit, E8200, 2.66G, 6MB, 1333FSB, 65W, C0
Wolfdale Dual Core Kit, E8300, 2.83G, 6MB, 1333FSB, 65W, C0
Wolfdale Dual Core Kit, E8400, 3.0G, 6MB, 1333FSB, 65W, C0
Wolfdale Dual Core Kit, E8500, 3.16G, 6MB, 1333FSB, 65W, C0
So you can go Quad up to Q9400 for sure or Q9650 most likely, however cooling might be a issue.
Another thing to consider. You said that your RAM is ONLY 4 GB, it is too small for today games. I play Battlefield 3 and I notice (I have 8 GB RAM 1333, my signature) that BF 3 takes about 4.2 GB RAM, so this could be your freezing issue.
So I would advise, first to upgrade your RAM, simple and cheap. Since Inspiron 560 is sibling to Inspiron 570 (mine) I strongly suspect that maximum RAM supported is 16 GB (Officially only 8 GB), so you can use 2 RAM sticks 4 GB each, and if you so desire, you can add 8 GB more later.
If RAM upgrade is not going to solve your issue, CPU upgrade would be next.
Regarding Video Card - 7750 is the strongest card Inspiron 560 300 watt PSU can support, if you want stronger, you would have to upgrade power supply.