What does a climax community lead to?
As pointed out by @bshane it seems your question is answered by the definition. From the relevant wikipedia articles


  "A seral community (or sere) is an intermediate stage found in ecological succession in an ecosystem advancing towards its climax community."


and 


  "In ecology, climax community, or climatic climax community, is a historic term that expressed a biological community of plants, animals, and fungi which, through the process of ecological succession the development of vegetation in an area over time, had reached a steady state. This equilibrium was thought to occur because the climax community is composed of species best adapted to average conditions in that area."


It doesn't lead anywhere, it is the climax. It is the state a community reaches when there is little change in the biodiversity as it because the community is composed of species at proportions that suit the ecology well. 

However, in reality it seems that climax communities are rarely (if ever obtained) and, with increasing climate change and habitat invasion by humans, few communities will be in equilibrium. The adaptive landscape varies in time and space, thus while a community may be well suited to the current environment, it may well be poorly suited to future incarnations of that environment and will be undergoing constant change. As such, the term "climax community" is somewhat obsolete with it generally referring to old or established communities rather than those at a climax.