In this paper, we study ${alpha}$-completely positive maps between locally $C^*$-algebras. As a generalization of a completely positive map, an ${alpha}$-completely positive map produces a Krein space with indefinite metric, which is useful for the study of massless or gauge fields. We construct a KSGNS type representation associated to an ${alpha}$-completely positive map of a locally $C^*$-algebra on a Krein locally $C^*$-module. Using this construction, we establish the Radon-Nikod$acute{y}$m type theorem for ${alpha}$-completely positive maps on locally $C^*$-algebras. As an application, we study an extremal problem in the partially ordered cone of ${alpha}$-completely positive maps on a locally $C^*$-algebra.