I know some general ways to build a matrix, which will have a given eigenvalues, but it will generally not be stochastic. For example, one can build canonical Jordan matrix by eigenvalues; yet it will be complex, if some of eigenvalues are complex, so it will not be stochastic. One can also use a generalization of it: a block-diagonal matrix, containing a block of this form for each complex eigenvalue pair $\alpha \pm i \beta$; yet it will have negative elements, so it will not be stochastic as well.
How to gain a stochastic matrix by eigenvalues?