(Maybe it shouldn't be an answer, as I won't say anything no one said before.)
A knot group is indeed a f.p. (obvious) group whose H1 is Z and H2 is 0 (by Alexander's duality). Moreover, as the meridian normally generates the group (that's a clear consequence of the Wirtinger's presentation), its weight is one. (The weight is the minimal number of "normal" generators. As far as I know, it's still a very mysterious group invariant.)
One of the early successes of surgery theory was to prove that these properties characterize knot groups of high dimension. (Michel Kervaire, Les nœuds de dimension supérieure (1965), available here.)
But, as always, the dimension three is more restrictive. For example, a knot exterior is a Haken manifold so Waldhausen's work imply a lot of restrictions. For a start, the word problem for this knot has to be solvable.
I'm sure that all the things we know about geometric decomposition and 3-manifold groups give various restrictions (for example, Stallings's fibration theorem imply that if you have an epimorphism from a knot group to a f.g. group whose kernel if f.g., then it comes from a fibration on the circle; in particular, the kernel is a surface group and the image is Z).
But I think that no definitive answer is known. Globally, 3-manifold groups are still mysterious (we still don't know if they are linear!) and they are just very particular quotients of link groups so I think our knowledge of them is still limited.