I think that if G is a simple complex Lie group, and N is its unipotent radical, then the natural map G -> G//N is not surjective. The categorical quotient G//N is the Spec of a ring which is the direct sum of each irreducible representation of G once (this follows from algebraic Peter-Weyl). This ring is graded by the positive Weyl chamber intersected with the weight lattice (its multi-proj is the flag variety). Consider the point defined by the ideal generated by all non-trivial representations. This isn't in the image of G (the group element above it would have to send all highest weight vectors to 0, which is impossible since G acts invertibly), but is a perfectly good element of G//N.
Though now that I think about it, this seems to suggest that the categorical quotient isn't the coequalizer in the category of schemes, but just in the category of affine schemes. Which may be what's confusing you: sometimes you have to add more points to affinize the coequalizer of two maps between affine schemes.