To answer the original question, a non-singular complex projective variety has a canonical Kaehler manifold structure (given by the pullback of the Fubini--Study form and the standard complex structure on $\mathbb{C}P^n$). In the converse direction, a compact Kaehler manifold can be embedded into $\mathbb{C}P^n$ for some $n>0$ if the cohomology class of the Kaehler form is integral (Kodaira embedding theorem).

An interesting question is: what compact complex manifolds admit Kaehler structure? A review of some obstructions to Kaehlerness can be found here. Let's list some notable obstructions for complex manifold $M$ of real dimension $2n$ to be Kaehler

- the underlying smooth manifold of $M$ must admit symplectic structure
- odd Betti numbers $b_{2i+1}(M)$ should be even.
- there must exist a cohomology class $[\omega]\in H^2(X, \mathbb{R})$ such that wedge product $[\omega]^{\wedge j}:H^{n-j}(M, \mathbb{R})\rightarrow H^{n+j}(M, \mathbb{R})$ induces isomorphism
- the DG algebra of real $C^{\infty}$-differential forms on $M$ must be formal.

There is also a number of restrictions on the fundamental group $\pi_1(M)$ (see the book 'Fundamental Groups of Compact Kähler Manifolds' for nice exposition). One should note though that the requirements listed here are very far from being sufficient; apparently, we are still very far from having a complete characterization of compact complex manifolds admitting compatible Kaehler structure.