There is quite some work done exactly along the lines you suggest. Two sources that answer your question in somewhat different ways are:
Flagg's "Quantales and continuity spaces" and
Heckmann's "Similarity, Topology and Uniformity".
In more detail, given a quantale (that is a complete lattice together with a binary operation (taken to be commutative, associative, and whose unit is the bottom element of the lattice)) one can define a V space to be a category enriched in V. That amounts to a set $X$ and a function $d:X\times X\to V$ satisfying $d(x,x)\ge \bot$ and $d(x,z)\le d(x,y)+d(y,z)$. In that setting one can do quite a lot of the usual constructions, perhaps most notably the interpretation of Cauchy completeness (see http://ncatlab.org/nlab/show/Cauchy+complete+category). Particular choices of $V$ will recover familiar cases such as: ordinary (non-symmetric) metric spaces, posets, probabilistic metric spaces, as well as Lawvere's fundamental work on generalized metric spaces.
With any V-space one can associate (two) topologies in a way that extends the familiar one from ordinary metric spaces. Varying $V$ yields in this way familiar classes of topologies, most notably the Scott topology. This latter observation explains why V-spaces are studied in Domain Theory.
A nice early result is due to Flagg (and is related to earlier result of Kopperman): Every topological space is V-metrizable if one is allowed to choose $V$ based on the given topology.