Once you get the relay sorted out make certain the coils on the condenser are quite clean and unrestricted and the fan moves air properly. A frosted evaporator on a refrigerator
can also be a sign of a restricted condenser.
There should be a tag or label somewhere stating what refrigerant and how much of it is needed. It looks like you already have a service fitting, the vertical tube with the black hex nut cap which is at 1:00 on your lower picture. The valve underneath will most likely show you what sort of refrigerant you use, but to be sure to RTFL(abel).
Likewise, if you clean off the relay you should find a part number there as well. The concern I would have there is there are a few different relays that look identical externally but have slightly different components. A part number should be easy enough to cross reference to an appropriate relay.
If you are going through this work, I'd strongly suggest replacing that inline filter or receiver/drier. If your system has a replaceable orifice, now is the time to at least inspect it. If you see a long copper tube the size of a piece of spaghetti, you will not have a replaceable one FYI. Since you are this far into it, if you can evacuate the system and then pull a vacuum on it for an hour that will also help by removing any residual moisture in the system too.
In regards to actually adding refrigerant to a system like this with only one service port, the usual procedure is to evacuate the system and then add the number of ounces noted on the tag, wherever it is.
Hope that helps and didn't overcomplicate anything, but like they say with airplanes "10,000 bolts to hold it together, but only one to take it apart!" and unless you cover all your bases you might miss the issue causing the problem.
Good luck in your endeavor!