Given a smooth projective surface, I consider vector bundles of rank $r$ with fixed Harder-Narasimhan filtration. Does there exist a moduli space of such bundles? If yes, how is it constructed and which are its properties (dimension, smoothness....)?
your question is related to my earlier question Moduli of Extensions.
There are some obvious problems which occure already if you fix the HN-factors themselves: Let $E,F$ be sheaves with Ext^1(E,F) one-dimensional. Then there are two isomorphism classes of extensions $0 \rightarrow F \rightarrow G \rightarrow E \rightarrow 0$. One is trivial, the other one is not. Moreover the trivial extension is the limit of a iso-trivial family of non-trivial extensions. Therefore the moduli-space cannot be separated.
Nevertheless, you can construct the moduli space of sheaves with HN-factors as an Artin-Stack: As pointed out by Arend Bayer, Bridgeland's Introduction to Hall-algebras (arXiv:1002.4372) is a good reference.