There is a theorem of Rosenlicht ("Some basic theorems on algebraic groups", 1956, Theorem 13) asserting that a quotient of a connected algebraic group by its center is linear. So a connected algebraic group with trivial center is linear.

Is it true of connected complex Lie groups? I.e. is a connected complex Lie group with a trivial center a subgroup of $GL(n,\mathbb{C})$? Is it algebraic?

non-affineexamples such as abelian varieties. But a "complex Lie group" is more narrowly defined. So your last formulation of the question is unclear. Early in Chevalley's treatment of affine algebraic groups he shows that such a group is linear, whereas some familiar real Lie groups are not. The detailed structure/classification shows that complex semisimple Lie groups are indeed linear, but for solvable Lie groups you'd have to look further into Hochschild's work including his old bookStructure of Lie Groups, etc. – Jim Humphreys Sep 21 '11 at 13:53