Yeah, everything's always more complicated than your first glance.

1. First of all, you're talking about a single speed. A fixed gear means that you can't coast.

2. You have two issues to resolve - chain line and chain slack.

Chain line is easy. The rear cog that you use has to line up with your front chainring. There's two different possibilities. If you really have a cassette, it's possible to remove all of the cogs and then replace all but the one you want to keep with spacers. The trick is how many spacers you use on each side of the cog so that it lines up perfectly with your front chainring. If you have a freewheel (my bet) the cogs are attached to a separate ratchet mechanism. Remove that and replace it with a single speed freewheel. Then you have to figure out how to spacer the front chainring to match the plane of the freewheel.

Chain slack take up. If you don't keep your chain pretty tight it will slap around a lot and occasionally jump off it's sprockets. Since your bike has vertical dropouts you can't simply pull the rear wheel back until the chain slack is tight. A complicating factor is that generally you can only adjust chain length in 2" increments. The easiest way to do that is to use a derailleur to take up the chain slack. There are other gizmos you can buy that do the same thing. A second method is to use a half link. If you look at a bicycle chain you'll see that it has wide links and narrow links. A half link is wide on one end and narrow on the other. It's advantage is that it allows you to adjust your chain length by a single inch. A third method is called a "magic gear". That's a combination of rear cog and front chainwheel that happen to require the exact length of chain that fits between your crankset and vertical dropout.

That's the short version. The devil is in the details and there's lots of details to everything that I've said.