Capitol State Forest is managed by the Washington State Department of Natural Resources (DNR) for many different activities. Motorized vehicles are allowed on certain trails here, as well as mountain bikers and horseback riders. Some trails are shared between user groups, while others are restricted to just one or a few. On any given year there may be some active logging happening as well. A map and the most up-to-date trail conditions from DNR will be indispensable in planning a route and knowing which trails are best for you.
Generally the north of the forest is more for motorized use, whereas the southern half of the forest has trails for mountain bikers, horseback riders, and hikers. Capitol Peak is in the middle, so you'll probably approach from the south.
There are multiple entrances and access points and a vast network of trails here, so there are many different routes to the summit. A popular and direct one is via the Sherman Road. Once you get to the top you will see several radio towers as well as sweeping views.