After purchasing an aircraft, how do I put the transponder in my name?
I assume you have a Mode S transponder. A Mode S transponder uses a unique 24 bit address assigned by the authorities to your aircraft to communicate with Mode S radars. If you do not have a Mode S transponder there is nothing that links your transponder to your aircraft registration.

If you purchase an aircraft you need to register it with the Civil Aviation Authority in your country. With that registration you will get a 24 bit address (in some countries you will always get it, in others you need to indicate that you have a Mode S transponder). The 24 bit address is usually given as a hexadecimal string of six characters, some countries use a 8 character octal representation. This code needs to be programmed in your transponder. It may be that the 24 bit address is the same as it was before. In that case nothing has to be done on the transponder.