You should read $c_1(X)<0$ as saying that the first Chern class of $T_X$ is negative, or the line bundle $K_X$ is positive, in the sense of curvature. But positive line bundles correspond to ample line bundles. This correspondence is sometimes called the Kodaira Embedding Theorem. See for example p. 181 of Griffiths-Harris, Principles of Algebraic Geometry.

You should read $c_1(X)<0$ as saying that the first Chern class of $T_X$ is negative, or the line bundle $K_X$ is positive, in the sense of curvature. But positive line bundles are ample line bundles. This fact is sometimes called the Kodaira Embedding Theorem. See for example p. 181 of Griffiths-Harris, Principles of Algebraic Geometry.