A Lie subalgebra of $\mathfrak{gl}(n,k)$ which is the Lie algebra of an algebraic subgroup of $GL(n,k)$ is called an algebraic subalgebra. Apparently there are Lie subalgebras which are not algebraic, even in characteristic zero. If $\mathfrak{g}$ is the Lie algebra of an affine algebraic group then it must be ad-algebraic, ie. its image in $\operatorname{End}(\mathfrak{g})$ under the adjoint representation must be an algebraic subalgebra. An example of a non-ad-algebraic Lie algebra is given on pg. 385 of [Lie Algebras and Algebraic Groups][1], by Tauvel and Yu. 


  [1]: http://books.google.com/books?id=ntKhAutD8I0C&pg=PA385&lpg=PA385&dq=Lie+Algebras+and+Algebraic+Groups,+by+Tauvel+and+Yu.&source=bl&ots=4V46-zVx1n&sig=Wiq7bS5rJocA-k09uJVYWXDdqTw&hl=en&ei=ZHbXSvmiG4HysgOx2cCLBg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CA4Q6AEwAA#v=onepage&q=&f=false