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 (negative) is that the Ricci curvature is positive (negative). See Aubin's book [Nonlinear Analysis on Manifolds: Monge-Ampere Equations][1]. A sufficient condition for the first Chern class being nonnegative is that you find a holomorphic section of the anticanonical bundle which is nonzero on a dense open set. A necessary condition for the first Chern class to be positive (negative) is that the integral of the Ricci form is positive (negative) over all compact complex curves in the manifold.


  [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