Is there an algebraic characterization of convergence spaces similar to Barr's characterization of topological spaces as lax algebras for the ultrafilter monad? I'm also curious about the same question for uniform and Cauchy convergence spaces.

$\begingroup$ Nice question! This has been bugging me for a long time :) $\endgroup$ – goblin Feb 11 '17 at 15:41
This is sort of an extended comment.
If you're interested in this sort of thing, you should be aware that there's a whole cottage industry centered around generalizations of Barr's characterization which apply to uniform spaces, approach spaces, and other "topological" categories. See, for example, Clementino, Hofmann, and Tholen. The basic setup is that you have a monad on the category of Sets, you extend this to (a generalization of) the category of relations, and then you define lax algebras for the monad. A lot of work goes into seeing which aspects of topology can be generalized to this setting. I can't resist mentioning that this story fits into a common abstract setup with the notion of a multicategory  see Cruttwell and Shulman.
Note that the axioms for a convergence space are all phrased in terms of convergence to a single point $x$. So in a Barrlike axiomatization of convergence spaces, one expects there will be no analog of Barr's lax associativity axiom, which relates convergence at many different points to convergence at a single other point. In fact, if you simply drop the lax associativity axiom from Barr's characterization, (so you just have a relation from ultrafilters to the original set which is laxunital) you get pseudotopoogical spaces, which can be identified as a certain subcategory of convergence spaces.
To get convergence spaces you would presumably have to work with the filter monad rather than the ultrafilter monad. Be aware that there is more than one reasonable way to extend the filter monad from $\mathsf{Set}$ to $\mathsf{Rel}$  see Seal, which discusses two important ones, and Schubert and Seal, which discusses them more generally.