Like with any popular destination, the best time to visit Paris is off peak season. In case of the French capital it is after Easter (usually May), June and September, when prices of apartments in Paris are not yet skyrocketing.
Generally speaking, tourist crowds attack Paris the whole summer, but in July and especially in August it gets too hot and uncomfortable in the city. Further, many Parisians take vacations in August, which used to present a problem for visitors confronted with closed stores and temporarily shut restaurants. These days enough of Paris stays open to make your stay pleasant if it weren't for stuffy weather, so May, June and September are your best bets for a memorable visit.
However, because the French have 11 national holidays and the lion share of them falls on May, it will be wise to check museums, galleries, restaurants and stores you want to visit in advance. Calling is a good idea, and some of the places of interest just might remain open.
Paris is quite enjoyable in winter as well, especially midwinter, with Christmas bringing holiday buzz and, alas, higher prices, but numerous activities make it worth a trip.
Main advantages of renting an apartment in Paris
Long-term apartment rentals in Paris