There are very pedagogical proofs in [1] section 3.5 and in [2] section 6.1. In both references, the most typical properties of orthogonal subgroups of **finite-abelian** groups are proven, including the one you are looking for. I have read both texts and used both for preparing seminars, and I find the exposition quite clear. I have cited both of these references in one of my papers: I think the first of them [1] has never appeared in a Journal, but it is quite a trusted paper in my field (quantum computation), so you could cite the arXiv; the second reference [2] has recently been published in QIC Vol. 13. No.11&12, 1007 (2013).

In fact, this property is one of many consequences of the Pontryagin-Van Kampen duality, and holds for all **locally compact Abelian groups** (finite Abelian groups are a particular case of those; the integers, the real numbers, the torus are also locally compact Abelian; see also this **related question** of mine in MathOverflow). If you would like to see a proof of your statement in the more general case, I recommend you to take a look at the online notes of the course *Introduction to Topological Groups*, by Dikran Dikranjan, University of Udine. The relevant sections for you are:

- Section 11
*Pontryagin-van Kampen duality*, in particular 11.4.;
- the
*annihilator* $A_{\widehat{G}}(H)$, defined at the beginning of section 11.4.2, is precisely the subgroup of the characters of $G$ that you are looking at;
- theorem 11.5.4. proves your statement.

As far as I know, Dikranjan's notes have never been published; still, they are (in my opinion) a very useful resource. Some parts of them (but sadly not the ones you need) have appeared in *An elementary approach to Haar integration and Pontryagin duality in locally compact abelian groups*, Dikranjan and Stoyanov, Topology and its Applications, Vol. 158, no. 15, 2011 Elsevier Science.