Here is a very naive approach: choose a basepoint in the manifold (call it $M$). Then evaluation at the basepoint gives a map $$ \text{Diff}(M) \to M $$ and so cohomology classes on $M$ pull back to ones on $\text{Diff}(M)$.
If for example, $M$ admits a nowhere zero vector field, then using the associated flow one can construct a section of the above map, so cohomology classes in $M$ inject into the cohomology of the diffeomorphism group. This would seem to do what your asking, right?

Let me remark that the general problem constructing non-trivial cohomology classes in the diffeomorphism group is almost 50 years old and has a lot to do with higher algebraic K-theory. The early work of Hatcher and Wagoner, Hsiang et. al., Waldhausen, Igusa, Goodwillie, Weiss and Williams are some of the names that deserve to be cited in this context.

Here is a very naive approach: choose a basepoint in the manifold (call it $M$). Then evaluation at the basepoint gives a map $$ \text{Diff}(M) \to M $$ and so cohomology classes on $M$ pull back to ones on $\text{Diff}(M)$.

If for example, $M$ admits a nowhere zero vector field, then using the associated flow one can construct a section of the above map, so cohomology classes in $M$ inject into the cohomology of the diffeomorphism group. has the structure of a Lie group, then the above map has a section and the cohomology of $M%$ will inject into the cohomology of the diffeomorphism group via the section. 

This would seem to do what your asking, right?

Let me remark that the general problem constructing non-trivial cohomology classes in the diffeomorphism group is almost 50 years old and has a lot to do with higher algebraic K-theory. The early work of Hatcher and Wagoner, Hsiang et. al., Waldhausen, Igusa, Goodwillie, Weiss and Williams are some of the names that deserve to be cited in this context.