Impeachment would be a result of recognized grievances done to the American people - so this petition would state those grievances done to the American people by this adminstration. It would be a suit filed in a court of law.
Here's an interesting overview:
Although a petition is only as meaningful as its response, the petitioning right allows blocs of public interests to form, harnessing voting power in ways that effect change. The right to petition allows citizens to focus government attention on unresolved ills; provide information to elected leaders about unpopular policies; expose misconduct, waste, corruption, and incompetence; and vent popular frustrations without endangering the public order.
Yet the petition clause seems to strike most courts and legal commentators as obvious and uninteresting. Citizens and litigants invoke the First Amendment to secure Internet freedom, undisturbed worship, or a robust press, but petitioning rights don’t seem to attract much notice. The right to complain is hardly something that starts wars — or is it?
http://www.firstamendmentcenter.org/petition/overview.aspx