We propose a rate-filling method as a framework to study fast construction of bit-interleaved polar-coded modulation (BIPCM). In particular, the proposed method makes full use of the nesting structure of polar codes. We show that BIPCM can be constructed by appropriately allocating the rates of each polar subcode whose information set is easily extracted by a predetermined index sequence, e.g., the Polar sequence in the 5G standard. By this means, the reliability calculation and ranking operation of bit channels can be discarded, which is a desired feature for low-complex construction. Simulation results show the proposed fast construction method performs independently of the actual channel condition and is robust to diverse modulation and coding schemes in the 5G standard.