The first and, perhaps, most important step is know what you must prove. Each state has different laws and legal burdens. In many states, cohabitation does not, by itself, terminate a spousal support obligation. Often, only remarriage does so. As a result, you must demonstrate that a person cohabitating with your ex is contributing to her monthly expenses and, thus, reduces her need.
To prove that point, you would have to perform discovery (which is only possible when a motion is pending) or hire a private investigator. You could have a PI look through the garbage to see if mail arrives addressed to the co-habitant. A PI could take photos of the individual entering the residence and/or license numbers of vehicles and documenting when they are parked there. You could subpoena the landlord, you could acquire the checks used to pay the rent and determine who was paying it.