What does one generally do to file a wrongful termination suit in an at-will place of employment?

You are assuming your termination would be wrongful it would most likely not be, however. If you have a written employment contract and the employer violates it e.g. fires you when, under the terms of the contract, they may not, then you could sue for breach of contract in any county court. Or if you are discriminated against or harassed because of a specifically protected characteristric, the main ones of which are race, sex, religion, age over 40, or disability, you may have an employment discrimination lawsuit, which you could bring a complaint to the federal EEOC or the state equal/civil rights agency. But note this doesn't mean, for example, that a 50-year old African American Muslim disabled woman couldn't be fired--just that she can't be fried because she is 50, African American disabled, Muslim, etc. She can be fired for any other reason, including performance or a personality conflict with her boss.
Other than the above, at-will employment means you could be fired at any time, for any reason, and so termination would not be wrongful. Or your employer could simply make your work environment/job awful and hope you will quit--that is legal, too.