The first Chern class of a Kaehler manifold is the cohomology class of the Ricci form. A sufficient condition that the first Chern class is positive is that the Ricci curvature is positive, etc. See Aubin's book [Nonlinear Analysis on Manifolds: Monge-Ampere Equations][1]. Another sufficient condition is that you find a holomorphic section of the anticanonical bundle which is nonzero on a dense open set.


  [1]: http://www.google.ie/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=0CFwQFjAE&url=http%253A%252F%252Fbooks.google.com%252Fbooks%252Fabout%252FNonlinear_Analysis_on_Manifolds_Monge_Am.html%253Fid%253DNcLiTzEwFbcC&ei=1CXcT5exDIeZhQfb4ZCXCg&usg=AFQjCNFWV_PlxirqqHua2IZDJBXm1nBWKg&sig2=6JcmG_kKO40JAkrLwCwykQhttp%3A//www.google.ie/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=0CFwQFjAE&url=http%253A%252F%252Fbooks.google.com%252Fbooks%252Fabout%252FNonlinear_Analysis_on_Manifolds_Monge_Am.html%253Fid%253DNcLiTzEwFbcC&ei=1CXcT5exDIeZhQfb4ZCXCg&usg=AFQjCNFWV_PlxirqqHua2IZDJBXm1nBWKg&sig2=6JcmG_kKO40JAkrLwCwykQ