How can I make my client pay me for work I've already completed?
You can try to track down the company who owes you money, but if your agreement was with a company that was incorporated (or formed an LLC) and the company is now out of business, the likelihood that you will actually recover what you were owed is slim.
However, if you performed services for an individual who was simply using a company trade name, it may be worth going forward with a collection lawsuit as any judgment you get will be on the books (and collectible) for a number of years.
In addition, if you were really performing employee-type services (and were misclassified as an independent contractor), you may also have some recourse against the individual company owner for failure to pay wages.
It's best to confer with a litigation attorney who has experience in employment and collection law to assess the costs and likelihood of success, depending on your true situation.