The process is spelled out in the Administrative Procedure Act. See, e.g., National Urban League v. Ross and Ohio v. Coggins. Basically, it amounts to a lawsuit against a federal government official in the chain of command involved in making the decision in question filed in a suitable U.S. District Court.
The proper way to go about it is detailed by ohwilleke's answer. But for the sake of amusement let's ignore that and consider the fallback:
Reapportionment due to the Census, while required by the constitution, is not automatic and self-executing by default. It requires an act of Congress to determine both how many seats ...