I totally missed the coining of the term "Approximate Dynamic Programming" as did some others. Also, in my thesis I focused on specific issues (return predictability and mean variance optimality) so this might be far from complete. That's enough disclaiming.
Let's start with an old overview: Ralf Korn - Optimal Portfolios. Kenneth Judd - Numerical Methods in Economics gave me some good background on approximation of functions. You may not need it.
Brandt, M.W. et al. - "A Simulation Approach to Dynamic Portfolio Choice with an Application to Learning About Return Predictability" does what it says and uses value function approximation to do it. Van Binsbergen, J.H. and Brandt, M.W. - "Solving dynamic portfolio choice problems by recursing on optimized portfolio weights or on the value function?" improve upon this idea. Garlappi, L. and Skoulakis, G. - "Solving Consumption and Portfolio Choice Problems: The State Variable Decompostion Method" provide important numerical improvements.
The literature of Bertsekas should be of interest as his papers are often cited. However I haven't looked at them yet.
I hope this interests you, if this is not what you meant, do tell.