If you want a construction entirely compatible with Bushnell and Kutzko's theory of strata and simple characters (and that also works when $F$ has positive characteristic), you may refer to my PhD thesis :

Broussous, P. Extension du formalisme de Bushnell et Kutzko au cas d'une algèbre à division. (French) [Extension of the Bushnell-Kutzko formalism to the case of a division algebra] Proc. London Math. Soc. (3) 77 (1998), no. 2, 292–326.

For other reductive groups, there are basically two "schools". First Bushnell and Kutzko (GL(N), SL(N)) and the students of Bushnell (Shaun Stevens : classical groups), of Henniart (myself and Vincent Secherre : GL(m,D)), of Zink (Martin Grabitz : GL(m,D)).

(I don't give any precise references for you may easily find them with Mascinet.)

Second, you have the "american school", initiated by Roger Howe, it has entirely solved the construction of "tame" supercuspidal representations for a general reductive group. Howe itself did GL(n) a long time ago. The following papers solve the general case.

Yu, Jiu-Kang Construction of tame supercuspidal representations. J. Amer. Math. Soc. 14 (2001), no. 3, 579–622.

Kim, Ju-Lee Supercuspidal representations: an exhaustion theorem. J. Amer. Math. Soc. 20 (2007), no. 2, 273–320.

To finish I must add that Bushnell and Kutzko have defined the beautiful notion of "type" for Bernstein blocks of the category of smooth complex representations of a given reductive group :

Bushnell, Colin J.; Kutzko, Philip C. Smooth representations of reductive $p$-adic groups: structure theory via types. Proc. London Math. Soc. (3) 77 (1998), no. 3, 582–634.

This notion allows to develop a general strategy to construct all representations of a given reductive group.