Motivic vs Deligne cohomology
The existence of a cycle class map from motivic cohomology is a general fact to every cohomology satisfying certain axioms. For example, to every mixed Weil theory in the terminology of Cisinski-Déglise (see here, here the arXiv preprint) or more generally to every cohomology represented by a spectrum satisfying certain axioms (cf Theorem in the Introduction here, preprint here). As you see, for these results you need your cohomology to be given by a spectrum in Voevodsky's stable homotopy category. The Deligne cohomology is known to be given by a spectrum, thanks to Holmstrom-Scholbach in this paper (preprint here). The cycle class map and the Chern character for Deligne cohomology are defined in Definition 3.7 of Holmstrom-Scholbach.
If you are interested in the computation that such maps coincide with the classic definition, you should also check this paper by Riou (preprint here). In Definition 6.2.3.9. and Remark 6.2.3.10. Riou shows that the canonical map in spectra from $K$-theory spectrum to Beilinson's motivic cohomology spectrum composed with the canonical isomorphism to the Eilenberg-McLane spectrum with rational coefficients coincides with the classical Chern character.
In addition to the papers mentioned, I would like to add Kerr, Lewis, Müller-Stach, The Abel–Jacobi map for higher Chow groups, Compositio (2006). They give an explicit construction (if you like that sort of thing -- I do), using currents, mapping a cubical Bloch complex to a complex computing Deligne cohomology. There is also a follow up paper by the first two authors (with obvious title) in Inventiones.
Some alternative references, maybe more classical (and with cycles):
Section 12.3.3 of vol I of C. Voisin: Hodge theory and complex algebraic geometry. Cambridge studies in advanced mathematics 76. Cambridge University Press 2002.
Section 7 of H. Esnault and E. Viehweg: Deligne-Beilinson cohomology. In: Beilinson's conjectures on special L-values. Eds. M. Rapoport, P. Schneider, N. Schappacher. Perspectives in Math. 4, Academic Press 1988, pp. 43-91.