Upgrading Graphics Card on ASUS CM6870-US015S

Upgrading Graphics Card on ASUS CM6870-US015S

I want to upgrade my Graphics card(NVIDIA GT 620) on my computer(ASUS
CM6870-US015S) and I have a couple of questions.
I noticed that the PSU on my Graphics card is only 350W so I'm assuming
that I will have to upgrade that.
After a little searching I decided upon Corsair CX-750 with 600W and the
EVGA GeForce GTX760.
I was wondering if the above mentioned hardware will work well with my
comptuter and if there will be any issues with my motherboard or anything
else. I am aware that this will void my warranty but I just want a
powerful computer.
Am I missing anything so far? I have to add that I intend to use multiple
monitors at 1920x1200
Also would I have to add any cooling if I make this upgrade (Don't intend
to overclock just yet)
My System Specifications are
Processor: Intel Core i7-3770
GPU : Nvidia GT-620
Expansion Slots:2 x PCI-e x 1, 2 x PCI-e x 16
Power Supply: 350W
I couldn't find the exact motherboard model. The only thing I could find
in System Information is System Model: CM6870