Let $X$ be a finite complex. Then the functor
$$\lim_X:\operatorname{Fun}(X,\operatorname{Sp})\to \operatorname{Sp}$$
sending a local system of spectra $E$ to its limit preserves all colimits. Indeed it preserves all finite colimits by stability, and it preserves all filtered colimits by the finiteness of $X$. Therefore it is of the form
$$\lim_X E \cong \operatorname*{colim}_X(E\otimes \omega_X)$$
for a certain local system of spectra $\omega_X$ (this is by the universal property of the presheaf category and the fact that $\operatorname{Fun}(X,\operatorname{Sp})$ is the stabilization of the presheaf category). The local system $\omega_X$ is called the *dualizing sheaf* of $X$. Now if we let $\mathbb{S}_X$ be the constant local system at the sphere spectrum $\mathbb{S}$ we have
$$\mathbb{D}X_+=\operatorname{map}(\Sigma^\infty_+X,\mathbb{S})\cong\lim_X \mathbb{S}_X\cong \operatorname*{colim}_X \omega_X$$
where the right hand side is some sort of generalized Thom spectrum.

When $\omega_X$ is invertible (i.e. all its stalks are spheres), it is called the *Spivak normal fibration* of $X$, and $X$ is said to be a *Poincaré complex*. Note that $\omega_X$ is rather explicit: it follows formally from the definition that
$$\omega_X(x)=\lim_{z\in X^{op}} \Sigma^\infty_+ \operatorname{Map}_X(z,x)$$
where $\operatorname{Map}_X(z,x)$ is the space of paths from $z$ to $x$. Moreover one can show that for a closed topological manifold $X$ we have a natural equivalence
$$\omega_X(x)\cong \mathbb{D}\left(X/X\smallsetminus\{x\}\right)$$
where $X/X\smallsetminus\{x\}$ is the cofiber of the inclusion $X\smallsetminus\{x\}\subseteq X$.

This is equivalent to the construction explained in Gregory Arone's comment. A reference for this material is

J. R. Klein, *The dualizing spectrum of a topological group*, Mathematische Annalen 319 (2001), no. 3, 421–456, DOI 10.1007/PL00004441.

Another useful reference are the lecture notes for Jacob Lurie's class on *Algebraic L-theory and manifold topology*. In particular Lecture 26 is relevant.