What do you mean by your “Bitcoin Key”? You then talk about “Private Key”. Are you talking about different things?
How can I get to know my Private Key?
That’s what your wallet.dat contained. Obviously you cannot magically get that back if you’ve lost your wallet.dat. Otherwise, Bitcoin would be with $0 as anyone could just “make up” private keys with any amount of Bitcoin. Please use logical thinking.
I have no idea why you seem to think that installing Bitcoin Core would somehow retrieve a long-deleted wallet.dat.