For simplicity I want to stick to the orientable case. Further i want to assume, that $N$ has the structure of a CW-complex.

If $N$ is homotopy equivalent to a manifold $M$ then $N$ has to satisfy Poincare-Duality. For example, as said above, the dimension of $M$ is the dimension of the highest non-vanishing homology, which has to be $\mathbb{Z}$. 

So this is of course the first obstruction involving the (co-)homology and the $\cap$-product.
So suppose $N$ satisfies Poincare-duality. Then $N$ is a so called Poincare complex. 

The question whether a given Poincare complex is homotopy equivalent to a manifold (the other way round) is one classical problem in surgery theory.

There is a involved obstruction process coming, which is very roughly desribed in the [wikipedia][1]. More details can be found for example in the books mentioned there.

There should also be twisted versions of Poincare duality int the nonorientable case, which should also give a complete answer.

I do not know, whether the additional assumption, that the Poincare complex $N$ is indeed a manifold with boundary has any consequences for the surgery obstructions.

  [1]: http://en.wikipedia.org/wiki/Surgery_theory#The_surgery_approach