The Cut The Knot website has a good overview of this construction, which is known as the Problem of Apollonius. It can be broken down into several versions, where finding a circle tangent to three circles is the most general. Less general versions, such as for three tangent circles, or two circles and a point have shorter constructions. An even more general version, perhaps, is constructing a circle that has a given angle with the given circles.
One possible construction is Gergonne's Solution. This involves sub constructions which are listed at the bottom of the overview page.
The comments to your question also point to other discussions, such as circle tangent to three circles and Wikipedia.