Mechwarrior online is built with a four-core processor as the baseline. You can run on a dual-core but you will find your processors, regardless of their speed, nearly maxxed at 100% during game play. For dual-cores this may change once DX11 is implemented as DX11 allows much of the processing to be offloaded from the CPU to the GPU on your video card - but we do not have an ETA for DX11 yet.
So if you want to run smoothly with the game in its current state you are looking at a minimum dual-core processor system. This would require you to replace both your system board and processor (and unfortunately very likely your memory too).
If you are looking for less expensive components you may want to look in an AMD AM3+ processor solution - they have several quad and six-core processors AMD AM3+ CPUs for less than $100 on NewEgg.com. Then you would need a motherboard capable of supporting an AM3+ processor - there are many good brands I tend to favor ASUS, GIGABYTE and MSI.
If you can afford it the Sabretooth R2.0 system board is a good solid board:
http://www.newegg.co...N82E16813131877 otherwise I would suggest
http://www.newegg.co...N82E16813131872
For a processor try:
http://www.newegg.co...N82E16819103996
Unfortunately, an upgrade of this magnitude would likely show up other deficiencies in your system. On mine I went from a AMD dual-core to an AMD eight-core and had to replace the power supply because my old 500W PS could not handle it when the graphics card started drawing more power. Also you will probably find that even if your memory does work with the new system board it will prove to be a bottleneck because technology-wise it is much slower that the other components of your system.
I am sure others who use Intel-based systems will give you some good pointers for going with an Intel upgrade - I find the Core i7 processors to be too expensive for my tastes.