Raspberry Pi controlled lamp using a relay & mains procedure
Here is a good way to do this project without getting yourself messed up with playing with live mains wiring.


Acquire one of the many AC remote controlled outlet devices such as that pictured below. There are plenty of choices available for low prices on eBay or Amazon. Can be lower than price of the Raspberry Pi.





Plug the AC unit into your mains outlet and then plug your unmodified lamp into the front of the AC unit. 
The wireless remote FOB contains a battery operated transmitter circuit that can be hacked to let the RasPi trigger the lamp on and off. The nature of the hack interface will be a far better learning experience for you than risking life and limb screwing around with your own relay controlled AC wiring.
The simplest hack for the FOB will be to simply bring two small wires out of the FOB that connect to the active contacts of the push button and wire those over to your relay module. Everything is low voltage and safe.


Note that these low cost remote control units may not be of the highest quality but their usage will still be safer than what you have proposed in your posting.