We can produce Kirby diagrams for the complement of a ribbon surface $S$.
Indeed, there is a procedure that is described in *Gompf-Stipsicz "4-manifolds and Kirby calculus"*.
I'll briefly explain that, for more details see pg 211-213 of that book.

First we perturb the projection $B^4 \simeq \mathbb{D}^3\times I \to I$ to a Morse function for the embedded ribbon surface $S$. One can prove that a critical point of index $k$ for $S$ corresponds to attaching an handle of index $k+1$ to the complement $B^4\setminus \nu S$ (see Prop. 6.2.1).

Consider a diagram for the ribbon knot together with the bands specifying the ribbon moves. When we do band surgery along these bands we end up with a bunch unknots. The disks they bound corresponds to the 0-handles of $S$, while the bands are the (2-dimensional) 1-handles of $S$.
Now, in order to build the complement $B^4\setminus \nu S$ we start with a 0-handle and then attach a (4-dimensional) 1-handle for each disk above followed by a 0-framed 2-handle for each band. The attaching circle of the 2-handle has to follow the band (really it is a perturbation of the band such that it links the unknots (now appearing as dotted circles)).

I do not know what you can do for a generic slice surface since in principle you would have the extra complication of attaching 3-handles.
In the book is mentioned that it is not known any example handle decomposition for a topological slice disk which is not ribbon.