Predators are in the Serengeti all year. What changes is how easy they are to see and what they're doing, and those two things move with the grass.
Calving season, roughly late January to March
The predator peak, and not enough people know it. Half a million wildebeest calves are born on the short-grass plains of the southern Serengeti in a few weeks, which is the largest concentration of easy prey anywhere in Africa. Lions, cheetah, hyena and wild dog all converge. The grass is short, the ground is open, and you can see a hunt from a long way off. If big cats are your reason for coming, come now rather than in August.
The dry season, roughly June to October
The general-purpose answer. Grass is low and dry, water is concentrated, and animals are drawn to rivers and waterholes where they're predictable. Easy conditions for finding everything. It's also the busiest time, so popular sightings are shared.
Green season, roughly November to May
Harder work. Long grass hides cats, water is everywhere so animals disperse, and some tracks get difficult. In exchange: very few vehicles, lower prices, spectacular skies, and the best light of the year. Experienced safari-goers often prefer it and first-timers usually shouldn't start with it.
Species by species
Lion are reliable year-round across the Serengeti and the Ngorongoro Crater, with the crater the single most dependable place in the country.
Cheetah want open short grass, so the southern plains in calving season are the best bet by a wide margin.
Leopard are about habitat rather than season: the riverine woodland of the Seronera valley, in the trees, and it still comes down to luck and a guide who knows which trees.
Hyena are everywhere, far more interesting than their reputation, and a den with cubs is one of the better hours of a safari.
Wild dog are the rare one in the north; the southern parks are the place for them, and they get their own article.
Time of day beats time of year
Whatever month you come, the first two hours after dawn and the last two before dark are when cats move. A guide who gets you out at six, and an itinerary that doesn't have you driving between parks at dawn, will do more for your predator sightings than picking the perfect week.
Night drives, and where they happen
Night drives are where you see what actually comes out after dark: genets, civets, bushbabies, porcupines, and cats hunting rather than sleeping. They aren't permitted inside the Serengeti National Park, which surprises people. They are allowed in some private concessions on its edges and in parts of the southern parks, so if a night drive matters to you, that has to shape where you stay rather than being added later.