There are some relevant older results in section 5 of part II (on semisimple elements), of the 1968-69 IAS seminar notes by Springer and Steinberg included in Lecture Notes in Mathematics 131 (Springer, 1970). In particular, their Corollary 5.17 shows that a supersolvable subgroup of a semisimple algebraic group consisting of semisimple elements must lie in the normalizer of some maximal torus (defined over the finite field in your set-up). I'm [Here "supersolvable" generalizes the usual notion for a finite group by allowing subquotients to be either finite cyclic or algebraic tori.] I'm not sure whether there are any stronger results in the literature, but the discussion by Springer and Steinberg shows clearly that such questions are not straightforward to deal with in the algebraic group context.
As Marguax observes in a comment, being an $r'$-subgroup is the same as consisting entirely of semisimple elements.
Possibly more can be said in the context of finite groups of Lie type, using only finite group techniques, but it seems that the given subgroup $K$ consisting of semisimple (that is, $r'$-) elements must be of a rather special sort to embed in such a normalizer. ADDED: Keep in mind also that the normalizer of a maximal torus in the ambient algebraic group has a somewhat delicate structure in its own right, sometimes but not always a semidirect product of the torus and a copy of the Weyl group. (This has implications for the structure of the finite groups.) One standard reference is the paper by J. Tits, Normalisateurs de tores. I. Groupes de Coxeter etendus. J. Algebra 4 (1966), 96–116.
SUMMARY: In one direction, Springer-Steinberg give a uniform sufficient condition for a subgroup consisting of semisimple elements to embed in the normalizer: it's enough for it to be supersolvable. (This strengthens a classical result of Blichfeldt for finite nilpotent matrix groups.) In the other direction, there seems to be no useful necessary condition on the structure of such a subgroup of semisimple elements. Probably case-by-case study of the simple Lie types is needed to go further.
All of this is illustrated by comparing general linear and special linear groups. Here the Weyl group $S_n$ lives naturally in $\mathrm{GL}_n$ as the subgroup of permutation matrices; but only its rotation group $A_n$ consists of matrices of determinant 1. For large enough $r$ and $n$ this subgroup of the normalizer inside $\mathrm{SL}_n$ consists of $r'$-elements and is actually simple. Manguax's observation shows however that a finite nonabelian simple subgroup of $\mathrm{SL}_n$ (say over a finite field) may consist of semisimple elements but not lie in the normalizer. So there's unlikely to be any better general result than the one of Springer-Steinberg.