As Ryan Budney points out in a comment, there's not much you can say without specifying the slice disk $D$. If you have an explicit presentation of $D$ as a ribbon disk, then you can easily turn this into an explicit handle diagram for $D^4 \setminus D$. If you cut the ribbon self-intersections of the ribbon disk, the result is an unlink in $S^3$, and this unlink gives the 1-handles of the diagram (in Kirby notation, dotted circles). For each ribbon, add a 0-framed 2-handle which links each of the adjacent dotted circles.
As far as I know, every knot that is known to be (smoothly) slice is also known to be a ribbon knot (i.e. there are no candidate counterexamples to the "slice = ribbon" conjecture for knots).
If you have a slice disk $D$ which is not a ribbon disk, then the associated Morse function on $D$ will have local maxima, and each such maximum will correspond to a 3-handle in a generalization of the above recipe for a handle diagram.