CPU upgrade.

You need a desktop i7-3770 CPU using the LGA1155 socket.
They cost around 250€.

 

Maybe have a look for a teardown of your model then, iFixIt does not seem to have one yet. It will hardly be any different with the CPU socket.

 

You can void AppleCare doing this. You would have to get the original components back into the iMac before getting it serviced and hope they don't realize that you were inside re-arranging parts.

Only the memory in iMac is considered user serviceable, except for the current 21.5" which does not have user serviceable memory.

 

It will not work.

The 3.1 GHz CPU in your iMac is the i7 3770s. It runs at 65 watts. The CPU you're looking at is the i73770 you're looking at is the i7 3770. That CPU runs at 77 watts, which means it requires more power to operate in your system. Unfortunately, the power supply board in the 21.5" iMac does not supply enough power to run the 3.4 GHz CPU.

The 3770s is basically an underclocked 3770. I suspect that Apple chose the 3770s as opposed to the 3770 for the 21.5" iMac because it didn't produce as much heat.