I have just bought a bitcoin on coinbase, how do I get my private key?
you don't, at least not from coinbase...
I'd suggest you download a decent desktop wallet, generate an address in this wallet and then withdraw your coins from coinbase to your local wallet.
This way, you have controll over your own private keys, since you'll manage them with your desktop wallet. Do make sure you make a decent backup of your local wallet, since you'll be the one responsible for it