Hi Pauldro,
The Lenovo G560 laptop comes in a variety of configurations using a number of different processors, many come with i3 CPUs already installed. At least at this time most G560s appear to use the same motherboard which has an Intel G1 Socket aka the rPGA988A that fits a number of Intel Processors including the P6100, P6200, i3, i5 and i7. The G1/ 988 socket was designed for Mobile processors.
Not all versions of the P6100, P6200, i3, i5 and i7 processors fit into a G1 /988 Socket; some fit an 1156 desktop socket. You need to verify that you are ordering a processor with pins that fit the G1/988 Socket. Even then it is still possible you could end up with a version not compatible with the G560 motherboard. To be safe you should try to verify tha you are ordering an exact CPU that is currently or has been installed in a stock G560. This is more than just the type and the speed; you should try and verify the exact Intel model number.
The very fortunate thing about the G560 laptop is that the CPU is easily accessible from the back panel. We are fortunate also that the G560 and most other current laptops have socketed CPUs. There was a time when many if not most laptop CPUs were soldered in place. The CPU can be changed with very little effort. When you finally end up replacing the CPU it is very important to clean all of the residue off of the old heat suink compound and residue off the back of the fan and replace it with a good quality heat sink compound. You then need to verify that when the fan is tightened back down that it is making good contact with the back of the CPU. You also need to make sure you do a neat job and do not get excess heat sink compound in places where it could cause problems. Poor ventilation and the resulting heat is the number one killer of laptops.
You are going to need to do a little more homework before ordering a new processor. I have upgraded the processor in several laptops over the years. It has allowed me to extend the usefull lives of these computers. It has been very worth while in every case. It makes much more sense to wait until the CPUs drop in price. P6xxx, i3, i5, and i7 CPUs are all still pretty expensive right now. In a year or so the price will drop by half and there will be dozens of YouTube videos and posts in hardware hacker forums which will tell you exactly which processors will work best and how to do it. Depending on how you use your computer... currently you will probably get more bang for your buck by installing a faster hard drive and more memory.