how to legalize parking my new car in london?

Question

i live in zone 2, i presume i need to register somewhere to park a car on the street outside of the flat i rent. how do i do it?

and what costs are associated with that each month ?

Answer 1

There is no possible way for your question to be answered as we don't know where in zone 2 you live, as zone 2 is for rail travel.
You need to look a your neighbours cars to see if they have parking permits.
There would be bays marked on the road with signs that say "resident parking only"
If you have seen any of the above then you need to phone your council and apply. They will be able to tell you how much if at all.

Answer 2

Have a look at the residents parking disks on your neighbours' cars. The disks will tell you which council you should be contacting for your permit.