I have had the MW GOTY box set for PC for awhile. I now have a laptop which doesn't have a disc drive. How would I be able to install it and play it without being able to put the disc in?

You obviously cannot use the disc without a disc drive.

You could buy the game from Steam: http://store.steampowered.com/app/22320/

Or, for a little more money, you could buy a USB external disc drive which might be useful for other things as well.

Like this one:

http://www.newegg.com/Product/Product.aspx?Item=N82E16827151268

Or maybe someone you know has one you could borrow.