Descriptive model theory classification of Counting hierarchy
This is only a partial answer (to the $PSPACE$ characterization), but I don't have the reputation to comment. 

$PSPACE$ has the following (equivalent) descriptive characterizations:


$FO[2^{n^{O(1)}}]$, first-order logic with exponentially iterated quantifier blocks. 
$SO[n^{O(1)}]$, second-order logic with polynomially iterated quantifier blocks.
$SO[TC]$, second-order logic with a transitive closure operator.
$FO[PFP]$, first-order logic with a partial fixed point operator.
$CRAM$-$PROC[2^{n^{O(1)}}, n^{O(1)}]$, concurrent-read concurrent-write random access machine with exponential time and polynomial hardware.
$CRAM$-$PROC[n^{O(1)}, 2^{n^{O(1)}}]$, concurrent-read concurrent-write with polynomial time and exponential hardware. 


These are all mentioned in Immerman's book Descriptive Complexity.

I can't find a reference on a descriptive characterization for the counting hierarchy, but here are some observations:


$PP$ naturally corresponds to existential second-order logic with a second-order majority quantifier. 
$TC^0$ is equal to $FO$ with a majority quantifier. Padding arguments give us a relationship between $TC^0$ and $CH$. This relationship makes $SO$ with a majority quantifier (on relations) a natural candidate for a descriptive characterization of $CH$.