Looking for a good graphics card ( $150-275 )

Well, I need a new graphics card. Currently I have an AMD Radeon HD 6450.
I need something cheap.
I don't know much about PC's.
Right now I have a Pavilion HPE h8-1010, but I hope to make it into a gaming computer.
Everything is stock, so that's why I'm here.
Don't know if a new graphics card would even help.
Got my PC in 2011.
I can play LoL fine (60 fps or so), but for games like CoD my fps is terrible.
I need some reccomendations, and if I would need to upgrade anything else. (I've been told I need a new power supply.)

Specs

Processor type Intel Core i5
Processor speed 2.7 MHz
Graphics
Graphics Type Discrete
Secondary Graphics Chipset ATI Radeon HD
Ports and Connections
Available Interfaces HDMI
Memory
Memory size 8192.0 MB
Memory type DDR3
Memory speed 1333.0 MHz
Storage Media
Memory stick Memory Stick
Compact flash Yes
SD card

^ Copy and pasted this, because like I said, I'm not that smart with PC's

Suggest anything for under $300 please (I need good FPS & Just a better PC in general)

Also, I found this. (These games are pretty old lol)

Unreal Tournament 3 Medium 1024x768 61.1 Frames per second
Unreal Tournament 3 Medium 1680x1050 28.75 Frames per second
Unreal Tournament 3 Medium 1920x1200 22.67 Frames per second
Unreal Tournament 3 Highest 1024x768 46.29 Frames per second
Unreal Tournament 3 Highest 1680x1050 22.56 Frames per second
Unreal Tournament 3 Highest 1920x1200 18.68 Frames per second
Unreal Tournament 3 Highest 2560x1600 11.14 Frames per second
Far Cry 2, High Quality, 1024x768 16.6 Frames per second
Far Cry 2, High Quality, 1680x1050 10.0 Frames per second
Far Cry 2, High Quality, 1920x1200 8.4 Frames per second
Far Cry 2, High Quality, 2560x1600 5.3 Frames per second
Far Cry 2, Low Quality, 1024x768 36.4 Frames per second
Far Cry 2, Low Quality, 1680x1050 22.0 Frames per second
Far Cry 2, Low Quality, 1920x1200 18.6 Frames per second
Far Cry 2, Low Quality, 2560x1600 12.0 Frames per second
Dirt 2, High Quality, 1024x768 20.2 Frames per second
Dirt 2, High Quality, 1680x1050 13.1 Frames per second
Dirt 2, High Quality, 1920x1200 13.1 Frames per second
Dirt 2, High Quality, 2560x1600 13.1 Frames per second
Dirt 2, Low Quality, 1024x768 46.3 Frames per second
Dirt 2, Low Quality, 1680x1050 25.8 Frames per second
Dirt 2, Low Quality, 1920x1200 21.2 Frames per second
Dirt 2, Low Quality, 2560x1600 13.3 Frames per second
Dirt 2, Ultra Quality, 2560x1600 13.1 Frames per second
Far Cry 2, Ultra Quality, 2560x1600 4.8 Frames per second
Stalker 2, Ultra Quality, 2560x1600 2.0 Frames per second