EDIT: I have changed my mind. I think your best bet is the paperback Dover reprint by Bonola. He gives Bolyai's construction, which was slightly different from most modern treatments. Then he justifies it by nothing worse than the Law of Sines, which he writes as
$$ \sin A : \sin B : \sin C = \sinh a : \sinh b : \sinh c $$ Everything needed on pages 101-105 of Bonola.
ORIGINAL:Robin Hartshorne does this on pages 396-397 of Geometry:Euclid and Beyond with the proof that it works being a consequence of the formalism called Hilbert's Field of Ends. No hyperbolic trigonometric functions are used,and no animals are harmed. The bad part is getting people to believe in the field of ends (if you build it, they will come). I pointed out to Marvin Greenberg that the field of ends is simply the horizontal axis in a half-plane model. He really liked that, but i don't see that the notion dominates in the fourth edition of Marvin's book. However, Marvin showed in a 1979 paper that the real numbers are not necessary for validity of Bolyai's construction. In particular, any plane in which his "line-Circle" principle holds and the field of ends is Archimedean.
George E. Martin does this fairly quickly on page 437 of The Foundations of Geometry and the Non-Euclidean Plane, bu that is because he prepares with the trigonometry formulas first. On the plus side, Martin invented what Greenberg regards as possibly the only acceptable construction for reversing the process. The construction in the Dover reprint by Bonola would not work over a non-Archimedean field, for example.
Perhaps the quickest thing to get ahold of is Robert R. Curtis, Journal of Geometry, vol. 39 (1990), pages 38-59, Duplicating the Cube and Other Notes on Constructions in the Hyperbolic Plane.
Let's see, I emailed you a copy of Marvin Greenberg's article in the March 2010 Monthly. It is a little short on pictures. If you write back and let me know a little more about the course and the textbook I can probably help.
The most detailed books on constructions, by far, are in Russian from the late 1940's and early 1950's. I have photocopies of Nestorovich 1951, Smogorzhevsky 1951, both volumes of Kagan, 1949 and 1956.