If the compressor died from internal damage, it is not a do-it-yourself job unless the you is an air conditioner tech with the correct tools and parts.
When the compressor dies, it puts lots of crud and aluminum shavings in the system.
At a bare minimum, you will have to replace the filter/dryer, the orifice tube, flush out the lines, and maybe even put a suction line filter on.
Then the system will have to have a vacuum pulled on it for a while, and the correct amount of refrigerant and oil put back in.
Many years ago, the compressor went out on my Ford truck. I took it to a friend of mine, who has a full repair shop, and does things like this daily.
He advised me to take it to the local Ford shop.
At that time, the factory had a program that did all the above, and would give me a lifetime warranty on the compressor.
They had had a lot of returns on replacement compressors, and were trying to stop the bad reputation they were getting.
I had to pay a little extra for a suction line filter just ahead of the new compressor to catch any trash from the old one that would destroy it.
Plus, I had to agree to bring the truck back in after a certain period to get the filter replaced one time.
In return, they would pick up the whole tab for any new compressors that I needed.
I have never needed another one. They did it right the first time.
I doubt that that program is still ongoing, but wherever you take it, inquire about a warranty, and you might ask about a suction filter.
J.