The Gift of Death, chapter 1
Orgiastic religion is poorly defined in the chapter---perhaps Derrida is here relying heavily on Patocka, whom I have not read---and I've been unable to google up anything very useful. But to the extent that I understand the argument, the orgiastic rites from which Platonism emerge are enthusiastic---that is to say, initiates experience something like possession, an abandonment of self and volition to a communal experience of emotion and being. These orgiastic mysteries are then incorporated, but not subsumed or extinguished, by Platonism; Derrida draws on psychoanalytic meanings of "incorporation" that escape me, but, very broadly and in the signature deconstructive gesture, Platonism is shown to retain within itself secret traces of the orgiastic mystery it purports to surpass. Platonism introduces the responsible self at the threshold of the cave, that is to say, Platonic responsibility resides in one's vision of the Good. That gaze originates, crucially, from an internal and internalized seat of the self; it is a major objective of the chapter to show that this gathering in or "secreting" of the internal self and a kind of care or solicitude for death are mutually constitutive.
On to Christianity. Derrida argues that Christianity represses Platonism, again drawing on psychoanalytic meanings that I'm not equipped to evaluate, but again suggesting, very broadly, that Christianity retains secret traces---nay, broad swaths---of Platonism precisely as it denies those traces. The locus of Christian responsibility is the what Derrida calls the mysterium tremendum: the assymetrical relationship that opens the subject to the gaze of God but conceals God from the subject's own gaze. (It is this assymetrical nature of the gaze, I take it, that distinguishes Christianity from Platonism.)
Derrida goes on to make a further assertion about the modern---that is, Christian---self, namely, that the essential condition of Christian responsibility is heresy and dissent. The reasoning, if I've got it, goes as follows:
1. Responsibility can never know itself---that is, cannot acknowledge its historical origins, its historicity---but must claim for itself an autonomous, ahistorical transcendence
2. Because responsibility lacks self-knowledge, it can never act on the basis of comprehensive knowledge of the world
3. Thus responsibility must always be self-authorizing and self-authenticating and self-legitimizing, because it cannot legitimize its actions and choices on the basis of complete knowledge
4. That is to say, because it is self-authorizing rather than subject to a higher authority, responsibility is always already heretical.
I will leave it in the able hands of commenters to tease out the implications of this chapter for our project; I'm still in the digesting stage. Actually, still in the eating stage; by all means please correct my misapprehensions, of which I have no doubt there are many.