I am probably confused. Your h seems always to be surjective. What if C = Y, f = id, and C = X, and phi = constant map to point p of C? Then we seem to need h to be constant. E.g. taking h = phi, and g = id, seems to work (and the minimum degree which occurs here is zero). Is it possible your construction (essentially) works as long as the images of f and phi meet? I.e. that it works more widely, but that it is only when the image of f is actually contained in the image of phi that h is surjective? No I guess you need the inverse image of f(C) in X, i.e. the fiber product, to contain a curve, in order for a non constant g to exist. Something like that?
Indeed it seems more complicated than that. What if we take f the inclusion of a plane curve C into P^2 = Y, and X the blow up of P^2 at a point p of C. Then the we seem to be able to find both constant and non constant models for h. I.e. C' = C, and either h = id, and g is the "proper transform" of f, or h is a constant map from C to the point p, and g is any non constant map to the exceptional P^1 in X.
So there seem to be various answers of various degrees corresponding to different positive dimensional components of the fiber product, and whether or not they surject onto f(C). Indeed if there is a single point f(p) on f(C), whose inverse image in X contains a curve D, then we seem to be able to take C' = D, and g= id, and h = the constant map to p. Thus in general the minimum degree of h seems to be zero, e.g. whenever dimX > dimY and phi surjects. Does this seem right?