I have a copy of the wallet.dat file from the bitcoin-qt client that I had installed on an old computer. I know that that file contains my bitcoins.
Today I installed the client on my new computer because I want to use my bitcoins, but at the rate the sync is happening I think it will take a week or more to complete.
Is there any faster way that I can pull my bitcoins out of the wallet.dat file and use them?
You could create a wallet at blockchain.info by importing your wallet.dat file:
https://blockchain.info/wallet/import-walletor
You could dump the private keys from your wallet and import them into MultiBit.