UPDATE:
Ok, although the responses have been fine, I think I should clarify a bit further since the title is a bit too strict. I am not planning to run a game without death/dying, more I don't want combat to mean everything dies at 0 hp and I don't want it to be the goal of combat. Defeating your opponent is the goal, not killing them.
@vincegetorix has the right idea for the most part, and others are certainly there or on the right track as well.
Here is my thoughts for the system at present:
1.Your HP at level one is your Death Threshold. For creatures, it will be their HD size at maximum plus CON mod (or best ability modifier, haven't decided). If damage takes you to 0 hp and the remaining damage exceeds your Death Threshold, you are killed instantly. [I don't expect this to happen often, but with some attacks / spells if your hp is on the low side, it could happen.]
2. If you are reduced to 0 hp from a critical hit, you are unconscious and dying. You lose 1 hp per round until you die (passing your Death Threshold) or someone stabilizes you. [Note: critical hits in our game are when you roll maximum damage, not on a natural 20.]
3. If you are reduce to 0 hp from non-critical damage, you make a DC 15 CON save to remain conscious. If you remain conscious, you can move or take an action, no bonus actions or reactions though. You cannot attack or cast a spell. Once your hit points are restored to greater than your Death Threshold (or maybe half maximum hp?), you can act normally. [You're out of the fight at this point, until you get healed by a significant amount.]
Those are my initial thoughts. There will be no raise dead or revivify due to the nature of the campaign/setting. Creatures that are killed (other than old age) rise as undead, including PCs.