I intend to check the diagonal of L, where A = L'L, for negative elements. However, I don't know if Cholesky is meaningful in theoretical / computational sense if there are some negative eigenvalues.
Using the Cholesky decomposition is the quickest way I know of checking if a symmetric matrix has negative eigenvalues. Nothing wrong with that. Plus, if it succeeds, you already have the Cholesky decomposition! Of course, if there are any negative eigenvalues, it will fail. You will not need to check the entries of $L$.
Cholesky factorization is for symmetric positive definite matrices, and it will fail if the matrix has negative eigenvalues. You should use
singular value decomposition for that purpose, or maybe a QR algorithm would suffice if you just need some of the eigenvalues.
Edit 1: Of course, I should also add that, in general, L would not give you much information aside from what you would obtain through Sylvester's law of inertia.
Edit 2: I just thought that you may be calling LDLT algorithm, which is a generalization of Cholesky factorization, as the Cholesky factorization. In which case, you can use Sylvester's law of inertia to see if there are any negative eigenvalues.