A dual reordering strategy based on both threshold and graph reorderings is introduced to construct robust incomplete LU (ILU) factorization of indefinite matrices. The ILU matrix is constructed as a preconditioner for the original matrix to be used in a preconditioned iterative scheme. The matrix is first divided into two parts according to a threshold parameter to control diagonal dominance. The first part with large diagonal dominance is reordered using a graph based strategy, followed by an ILU factorization. A partial ILU factorization is applied to the second part to yield an approximate Schur complement matrix. The whole process is repeated on the Schur complement matrix and continues for a few times to yield a multilevel ILU factorization. Analyses are conducted to show how the Schur complement approach removes small diagonal elements of indefinite matrices and how the stability of the LU factor affects the quality of the preconditioner. Numerical results are used to compare the new preconditioning strategy with two popular ILU preconditioning techniques and a multilevel block ILUT preconditioner.