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Provided that the facets of $P$ have integral normal vectors, this is always the case. To construct such a tropical variety, first let $\Sigma$ be the normal fan to $P$. This comes along with a piecewise linear function $\psi$ defined by $\psi(n)=-inf\{ \langle n, m \rangle\,|\, m\in P\}$. Now choose any lattice polytope $Q$ containing the origin as an interior point, and restrict $\psi$ to $Q$. Take the tropical polynomial $$f=\sum_{m\in Q\cap {\bf Z}^n} \psi(m) z^m.$$ It is then a simple exercise in convex geometry to see that this tropical polynomial defines a tropical variety, which, when viewed as a polyhedral complex, contains the boundary of the polytope $P$. In particular, the PL function defined by this tropical polynomial is the Legendre transform of the function $\psi$. You can see some details of this in, say, Mikhalkin's paper http://arxiv.org/pdf/math/0312530.pdf, in section 3. If you want to have the polytope $P$ itself appearing, then you can take the tropical variety thought of as the graph of $f$, obtained by taking the tropical polynomial $y+f$, where $y$ is an extra variable: see Proposition 3.5 in the above cited paper.