NFS the Run use the grafic engine Frostbite 2 , used to develop BF3. I think you need a much powerfull video card. If you play on 1920x1080 resolution you need a 560 or 6950. But i think an 6870 will be fine. This are the requirements to run NFS the Run :
Need For Speed The Run Minimum System Requirements
CPU: 2.4 GHz Intel Core 2 Duo or equal AMD
RAM: 3 GB
GFX: 512 MB RAM ATI Radeon 4870 or better or 512 MB RAM NVIDIA GeForce 9800 GT or better
DX: DirectX 10
OS: Windows Vista SP2 32-bit
HDD: 18 GB
Audio: DirectX compatible
ODD: DVD Rom
Network: 512 KBPS Internet connection or better
Need For Speed The Run Recommended System Requirements
CPU: 3.0 GHz Intel Core 2 Quad or equal AMD
RAM: 4 GB
VGA: 1024 MB RAM ATI Radeon 6950 or 1024 MB RAM NVIDIA GeForce GTX560
DX: DirectX 11
OS: Windows 7 SP1 64-bit
HDD: 18 GB
Sound: DirectX compatible
ODD: DVD Rom
Network: 512 KBPS Internet connection or better