Setting up chain tension on a fixed gear with road bike frame

The short answer, is you generally cannot make a fixie out of a frame with vertical dropouts.

Not only do you need to tension the chain, but the spacing of the axle of the bottom bracket and rear wheel varies, depending on your selection of cog and chainring tooth counts.

Adjusting tooth counts will allow you to try to fine adjust the spacing, but it won't always get you there. The formula for how adding or removing a tooth will change your ideal chainstay length is not simple. You're removing a link from the chain, but you're removing it at an angle, so your reduction in chainstay length will not be the same as the amount of chain you remove.

Online, you can find many magic gear calculators [example] where you can try to pick a ratio that will exactly match your chainstay length.

You can also use a special half-link chain element, to adjust the chain by half as much as an ordinary link.

With all that said, the difference between "slack" and "tight" is a very small amount of chain stretch, and fixies in particular are pretty touchy for the tension. What I describe above I think works well for coaster-brakes and single speeds, but I would not recommend it for fixies.

If you are especially willing to throw caution to the wind, you could file the dropouts to be slightly wider, which might buy you just enough space to make one of the magic gear ratios work. The risks to safety should be obvious but that's up to you.

If you cannot control the rear cog's position, you can try moving the front chainring to tune the chain tension. That is, get an eccentric bottom bracket:

By rotating it in the frame, the distance between rear and front cogs can be tuned.

The same idea is achieved by eccentric rear hubs, e.g. White Ind. Eno:

The hub choices below offer those of you looking to convert a frame with vertical dropouts, a choice of eccentric hubs.