There is a paper of Larsen and Pink dating (Update: It has appeared: Larsen, Michael; Pink, Richard: Subgroups of algebraic groups. J. Amer. Math. Soc. 24 (2011), 1105–1158.)dating back to 1998 (but still in the process of getting published - long story there) that gives a conceptual proof (based on algebraic geometry methods, primarily) that any sufficiently large finite simple group that has a bounded rank linear model (i.e. it is isomorphic to a subgroup of $GL_d(k)$ for some field k and some bounded d) is basically of Lie type. So this, combined with the classification of simple groups of Lie type, gives an answer to the question in the bounded rank case. Unfortunately this isn't the whole story because one can certainly let the rank go to infinity, and then there are also the pesky alternating groups which are not of Lie type at all (except, perhaps, over the field of one element, whatever that means...).
There is a paper of Larsen and Pink dating back to 1998 (but still in the process of getting published - long story there) that gives a conceptual proof (based on algebraic geometry methods, primarily) that any sufficiently large finite simple group that has a bounded rank linear model (i.e. it is isomorphic to a subgroup of $GL_d(k)$ for some field k and some bounded d) is basically of Lie type. So this, combined with the classification of simple groups of Lie type, gives an answer to the question in the bounded rank case. Unfortunately this isn't the whole story because one can certainly let the rank go to infinity, and then there are also the pesky alternating groups which are not of Lie type at all (except, perhaps, over the field of one element, whatever that means...).