In southern Africa (Botswana, Namibia, Malawi, South Africa, Zambia and Zimbabwe) the best wildlife viewing is during the dry season period from May through October each year with July, August, and September being the most desirable months in terms of climate and temperature.
In East Africa (Kenya and Tanzania), exceptional wildlife viewing can be experienced during the dry seasons from January to March and June to October. On a typical year with normal weather patterns the migrating wildebeest and zebra herds will be in the Serengeti in Tanzania from December to June, in the Maasai Mara in Kenya from July to October and travelling between the two areas during the other months.
Wildlife viewing is, in part, predicated on the amount of surface water available. During the rains (December through March in southern Africa and April/May and November/December in East Africa) wildlife is scattered as there is abundant surface water to drink (puddles, waterholes, perennial rivers). Once the rains end wildlife congregates around remaining surface water (pools, lakes, and large rivers).
Please note that travel during southern Africa's 'green season' (November through April) and East Africa's short rain months (November and December) does have some advantages. The bush will be lush and green and many wildlife species will have newborn in tow. Dust, which can be a nuisance during the dry season, is greatly reduced and night-time temperatures are milder than during the dry winter months. Some camps such as Mombo in Botswana provide incredible wildlife viewing year round. Another Botswana camp, Duba Plains (which is renowned for its classic lion and buffalo confrontations), actually offers better wildlife viewing during the months October through May as the Okavango Delta flood waters are lower. The Victoria Falls are in full spate (heaviest flow) during the period January through June. Lastly, during the 'green season' many northern migratory birds visit Africa making the region an ornithologist's paradise!