Binary Black Hole Solution of General Relativity?

There are no exact solutions, only approximations and numerical solutions.

Don't forget that orbiting black holes will radiate gravitational waves so any solution would have to include those and the corresponding decay of the orbit until the black holes coalesce.

According to general relativity, a pair of massive bodies that orbit each other emits gravitational waves - for analogous reasons to the reasons why accelerating charges in electrodynamics emit electromagnetic waves.

So there can't be any static solutions resembling binary stars or binary black holes. The solutions have to be non-static and a complicated system of two orbiting bodies that emits gravitational waves - and eventually collapses into one object - clearly can't be solved analytically.

These things are usually discussed numerically, see other answers. In particular, the 1993 Nobel physics prize was given for an observation of a pulsar whose frequency changes in time exactly in the right way to be explained as the loss of energy caused by the emission of gravitational waves as predicted and calculated by general relativity.

For a very recent authoritative review of the numerical approach, see Centrella et. al. http://arxiv.org/abs/1010.5260
For the alternate parameterized post Newtonian approach, see Living Reviews of Relativity http://relativity.livingreviews.org/Articles/subject.html
and look for articles number 2007-2, 2006-4 and 2003-6.