1. Let's start with the word 'mortal' and remind ourselves of some obvious points. 'Mortal' is from the Latin mors, mortis meaning death. That which is mortal is either subject to death, or conducive to death, or in some way expressive of death. Thus when we say of a human being that he is mortal we do not mean that he is dead, but that he is subject to death. My being mortal is consistent with my being alive and kicking. Indeed, if I weren't alive I could not be said to be either mortal or immortal. Spark plugs are neither mortal nor immortal. Some will say of a car that it has 'died.' But that is a loose and metaphorical way of talking. Only that which was once alive can properly be said to have died.
2. Although 'mortal' applies to all living things, what interests us particularly is 'mortal' as a predicate of human beings. To be mortal in this sense is to be subject to death. But this phrase has at least two senses, one weak the other strong.
WEAK sense: X is mortal =df X is able to die, liable to die, has the potential to die. Mortality as posse mori.
STRONG sense: X is mortal =df X has to die, is subject to the necessity of dying, cannot evade death by any action of its own, is going to die, will die in the normal course of events. Mortality as necessitas moriendi.
Correspondingly, there are strong and weak senses of 'immortal':
STRONG sense: X is immortal =df X is not able to die.
WEAK sense: X is immortal =df X is able to die, but is kept alive forever by a factor distinct from X.
3. Let's run through some cases to illustrate the distinction. God is not mortal in either the weak or the strong sense. It is built into the divine nature (essence) that he cannot die. 'God is dead,' taken literally is nonsense. (Of course, that is not the way Nietzsche intended it to be taken; he was making a cultural point.) God is a necessary being, a being that exists in all possible worlds and at all times in those worlds containing time.
Your humble correspondent is mortal in both senses. Not only can I die, I must die, I cannot do anything to avoid eventually dying: I am subject to the necessitas moriendi. Cryogenics won't help for reasons I won't belabor at the moment. It is worth noting that, according to Christian doctrine, my having-to-die is a contingent attribute of me unlike my being-able-to-die. My having-to-die is punishment for original sin and is as contingent as that sin. My being-able-to-die, however, is grounded in my nature as a soul-body composite, and I am essentially such a composite. Thus it is not my nature to be immortal (in the strong sense), whence it follows that if I achieve immortality (in the weak sense) it is due to a supernatural gift: God freely grants me immortality; I don't have it apart from free divine donation. In this sense I am not naturally immortal: I am not immortal in virtue of my nature or essence the way God is.
Prelapsarian human beings are mortal in the WEAK sense but not in the STRONG. Unlike God, there is nothing in the nature (essence) of such beings to prevent them from dying if they should will to die. But if they do not will to die, God grants them unending life. Postlapsarian human beings, however, are mortal in both the weak and the strong senses.
4. Now what about Jesus Christ? He is one person in two natures; fully man and fully God. But all men are mortal in the weak sense: they can die. So Christ is mortal in the weak sense. But he is not mortal in the strong sense: he is not subject to the necessitas moriendi. He freely chose his death.
So is Christ a counterexample to 'All men are mortal'? It depends on what is meant by 'mortal.' Taken in the weak sense, Christ is not; taken in the strong sense, he is.
5. But there is still a Christological problem that wants solving. If Jesus Christ is God (or, to be precise, the Second Person of the Trinity), then JC is strongly immortal. But if JC is fully human, then he is not strongly immortal, but weakly immortal. How can one and the same person have contradictory attributes?