Dehn gave at least three solutions of the conjugacy problem for surface
groups, which can be found in my translation in the book *Papers on Group 
Theory and Topology* (Springer 1986), Papers 2, 4, and 5. 

The first is based on an idea of Poincaré: lifting a curve to the universal 
cover, which is the disk model of the hyperbolic plane, replacing it by the
hyperbolic straight line with the same endpoints, then projecting this line
back to the surface as a "geodesic representative" of the original curve. 
Two curves are free homotopic (and the corresponding group elements are
conjugate) if and only if they have the same geodesic representative.

This unpublished proof is conceptually simple, but it is not clear how to
determine whether two curves have the same geodesic representative. Dehn's
first published proof (Paper 4, 1912) worked out a way to do this, arriving in
a roundabout way at Dehn's algorithm.

Shortly afterward (Paper 5, still in 1912) Dehn noticed that the algorithm
follows easily from combinatorial properties of the tessellation of the
universal cover, and the hyperbolic metric is irrelevant. This is essentially 
the proof we use today.