First, do you know the difference between a fixie and a single speed? Let's get this out of the way first:
A "fixie" is a bike with a direct drive. When the wheels move- forward or backward- the pedals go with it. Screaming down a hill at 40mph? The pedals spin merrily along... and unless you are very strong it will be difficult to control the bike to slow your progress. Can't make it up the hill? Guess what, like it or not you are going to pedal backwards... a very scary proposition.
A "single speed" is one that has (you guessed it) one speed and either a freewheeling system or a coaster (back pedal) brake system.
Now, I strongly recommend that you borrow or rent a fixie and/or single speed before cannibalizing your bike. You may find out that you don't like it.
You can purchase single speed conversion kits for bikes currently equipped with derailleurs. Ultimately, the kit includes a single cog, a bunch of spacers, and a chain tensioner. You remove the rear derailleur and the cassette (or freewheel, if thats what you have) and replace them with the appropriate parts. They cost about $40, perhaps less, at most shops. If you choose to have the shop install the setup (recommended) then add another $40-50 labor.
30 years in the industry