The best time to visit Kenya depends on which of four things you care about: wildlife, weather, crowds, or price. They peak at different times, and no single month wins all four.
If you want the migration
The herds are generally in the Mara between July and October, and river crossings are possible in that window.

They are not scheduled events. The herds move on rain. A crossing can happen four times in a morning, or not at all for three days. Treat any guarantee of one with caution.
If you want the best weather
January and February are dry, clear and hot. Late June to October is dry and cooler.

The long rains fall from March to May. The short rains come in November and are much less disruptive, usually an afternoon downpour rather than a washout.
If you want the fewest people
March to May. The parks are green, the lodges are close to half price, and you will have sightings to yourself.

You will also get wet, and some minor roads become difficult. In the Mara that matters less than people expect, because the main tracks stay passable.
If you want the lowest price
The same window. Low-season rates in the Mara can be forty per cent below July. The resident game does not leave when the herds do, the lion, leopard, cheetah and elephant are all still there.
Month by month, briefly

January and February. Dry, hot, clear. Excellent game viewing, and good mountain views in Amboseli. Busy over New Year, then quieter.
March. The long rains begin, usually late in the month. Rates start dropping.

April and May. The wettest months. Green, empty and cheap. Some camps close.
June. The rains end. The country is still green and the crowds have not arrived.

July to October. Peak season. Herds in the Mara, dry weather, highest rates, most vehicles. Book nine to twelve months ahead.
November. Short rains. Brief, and a genuinely good-value month.

December. Dry again. Quiet early, then very busy from about the 20th.
So what is the best time to visit Kenya?
If you want one answer: late June to October. It is dry, the game is concentrated around water, and the herds are in the Mara.

If you want value instead, come in the green season and pay close to half.
Book earlier than you think
The dry-season camps on the Mara conservancies sell out nine to twelve months ahead, and the good guides go with them. If your dates are fixed by school holidays, treat that as a booking deadline rather than a suggestion. Our private camps guide covers which ones are worth the wait.

The Kenya Meteorological Department publishes seasonal forecasts at meteo.go.ke if you want to check a specific month against the long-term averages.