We design multilevel coding (MLC) schemes with low-density parity-check (LDPC) codes as component codes at each level. We develop a method to analyze the performance of an LDPC code at any level as the codeword length goes to infinity, even if the equivalent binary-input component channels are not symmetric. By joint optimization of code rates and degree distributions, the optimized irregular LDPC codes at each level are capable of achieving reliable transmission at signal-to-noise ratios (SNR) very close to the capacity of the additive white Gaussian noise (AWGN) channel as the codeword length tends to infinity. Simulation results show that the optimized LDPC codes also perform very well at moderate codeword lengths.