Here are some more purely group theoretical conditions. This is also not a complete answer, since it gives just some necessary conditions for certain groups to be linear.
Schur: Suppose that $G$ is a linear group, such that all elements have finite order. Then $G$ is finite.
Jordan, Dickson: Suppose that $G$ is a finite linear group. Then there exists an integer-valued function $\beta(n)$ such that G contains an abelian normal subgroup of finite index at most $\beta(n)$.
Malcev: Suppose that $G$ is a finitely-generated linear group. Then $G$ is residually finite. Furthermore, if $G$ is simple, then $G$ is finite.
Brauer-Feit: Suppose that $G$ is a periodic linear group. Then $G$ is abelian-by-finite.
Malcev: Suppose that $G$ is a solvable linear group of degree $n$. Then $G$ contains a triangularizable normal subgroup of finite index bounded by a function of $n$.