You should probably add reference request tag here. Anyway I think that everyone agrees that the modern approach to the circle method relays on spectral theory and Fourier analysis. PS I don't think one needs much more than the residue theorem in complex analysis to deal with the Hardy-Littlewood method in its complex formulation.
– AsafOct 7 '13 at 6:49

Vaughan's book has already been recommended. Let me add that Nathanson's first volume on additive number theory ("the classical bases") also treats some of the basic applications of the method and is decidedly gentler than Vaughan's book.
– so-called friend DonOct 7 '13 at 15:22

Another classic book is Davenport's Analytic methods for Diophantine equations and Diophantine inequalities, which was edited by Browning and republished recently in the Cambridge Mathematical Library.
– LuciaOct 7 '13 at 16:18

The book I would recommend is Vaughan's The Hardy-Littlewood Method (Cambridge Tracts in Mathematics). I don't remember it (or the subject in general) using much complex analysis - only the properties of the function $e^{ix}$, really....