How to teach neural network a policy for a board game using reinforcement learning?
You need to use some function approximation scheme. In addition, experience replay would be useful for two reasons: (1) you want to keep past memories (2) you need to decorrelate the way to teach your network.

Have a look at Deepmind's DQN on ATARI games. What you are describing is basically what they have solved. The paper is in their website: http://deepmind.com/dqn.html 

Mnih V, Kavukcuoglu K, Silver D, Rusu AA, Veness J, Bellemare MG, et al. Human-level control through deep reinforcement learning. Nature. 2015 Feb 25;518(7540):529–33. 

With respect to the network architecture, it will definitely require some experimentation. For an alternative, you can also have a look at HyperNEAT. They evolve the network topology:

Hausknecht M, Khandelwal P, Miikkulainen R, Stone P. HyperNEAT-GGP: A HyperNEAT-based Atari general game player. In: Proceedings of the 14th annual conference on Genetic and evolutionary computation p. 217–24. Available from: http://dl.acm.org/citation.cfm?id=2330195

For more strategic games. Maybe you can have a look at "Giraffe: Using Deep Reinforcement Learning to Play Chess" http://arxiv.org/abs/1509.01549