If you don’t like hot weather, do not go to Doha in the summer.  Starting as early as May, the temperatures are going to break triple digits and will do so with increasing frequency until around September.  June, July, and August highs typically top 100, and temps above 110 are not uncommon in the summer.  

Winters, however, are milder, with sixties as common as eighties.  Maybe an inch of rainfall will occur outside the summer months, but no drops will fall basically from late April into early October.  Arid in the winter, humid in the summer, those prone to heat stroke are advised to stay in or to stay away.

Climate aside, there is no best time of year to stop in Doha, unless you plan on doing some golfing or touring the city on foot.  The worthwhile museums are always open and, as a city with minimal nightlife and festival calendar, a schedule of Qatari events and activities should not impact your itinerary .