The ethics in Kierkegaard is not defined through Christianity. Christianity is a given for Kierkegaard, what is important is the work he did on the kind of subjectivity necessary for grasping the 'Christian message'. Despite Kierkegaard's apparent commitment to militant Biblical Christianiy, his approach is an Enlightnment one in which the historical claims of the Bible are interpreted as spiritual communication in which the historical content does not need to be true. This is particularly evident in Philosophical Fragments. The point is to appeal to subjectivity and that is done through a variety of strategies. The most obvious strategies are those of ironic aesthetic reflection and the construction of a theory of subjectivity.
For Kierkegaard, subjectivity is freedom. The free reflection of subjectivity on itself properly understood leads us to an an ethics beyond the metaphysical in Concept of Anxiety. Metaphysics can only refer to a fixity of subjectivity which denies its freedom. Kierkegaard does refer to the second ethics as concerned with (Christian) doctrine. Nevertheless, Christian doctrine in Kierkegaard is a commitment to subjectivity which is grasped through ironic reflection, psychological theory, epistemological scepticism of a pre-Cartesian antique kind, and the paradoxes of the relation of subjectivity with the absolute. That can be grasped as the relation between the succeeding empirical stages of the self and the continuity of the self over time