There are plenty of sources discussing dual abelian varieties, but I'm looking for a reference that discusses the construction and properties of the dual abelian scheme. I'm willing to accept general theory that the Picard scheme of an abelian scheme exists, but why is it an abelian scheme?
You can also look at chapter 6 of Mumford's "Geometric Invariant Theory", together with part of Kleiman's article in "FGA Explained" (section 9.6) for the construction of $Pic_{X/S}^\tau$ from $Pic_{X/S}$. 


Alternately, see Faltings, Chai, "Degeneration of Abelian Varieties" Chapter I, especially Theorem 1.9 The general idea is: it can be shown that the Picard functor of a scheme $X/S$ is represented by a group algebraic space over $S$. If $X$ is an abelian scheme, then it can be shown that $Pic^0(X/S)$ is an abelian algebraic space. A Theorem of Raynaud shows that any such algebraic space is automatically a scheme. I think I recall that BLR only proves this fact for certain types of schemes $S$. 


See Bosch, Luetkebohmert, Raynaud, "Néron Models", chap. 8, 8.4, p. 234. 

