**EDIT:**
The content below and at the comments rely on Kahler manifolds, and this does not intend to be a complete or even satisfactory answer:

The case when the first Chern class is positive is rather delicate. It was disproved by Yau that even provided the manifold possess positive first Chern class, then there is no ensure for example that the manifold admits a positive Kähler Einstein metric. Also, even when the Kähler–Einstein metric exists, it need not be unique.

Further, a necessary condition for the existence of a Kähler–Einstein metric is that the Lie algebra of holomorphic vector fields is reductive. It was conjectured by Yau that provided the first Chern class is positive, a Kähler manifold has a Kähler–Einstein metric if and only if it is stable in the sense of geometric invariant theory.

More nice comments appear here: First chern class

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A necessary condition (assuming Kähler) to this rely on the fact the first Chern class of manifold is positive. Since the Ricci curvature of an hermitian metric is connected with its Ricci form, that determines the first Chern class.

On the otherside, provided $c_1(M) > 0$ and the manifold is compact and Kähler, then by the Calabi-Yau theorem your requirement is true.