I was given an actual casascius bitcoin minted in 2011 as a tip, and I'd like to know where I can exchange it for US dollars.
How would I extract the private key from this coin to gain access to the digital bitcoins if I were to sell them online?
You have two routes here:
Don't remove the sticker on the back. Use a website like localbitcoins to sell it to someone near you.
Some people might want a physical Bitcoin, especially since they're not being made anymore.
The remaining brass part is mostly worthless.