My player is playing a Death Domain cleric. They have the Channel Divinity option Touch of Death (DMG, p. 97), which deals extra necrotic damage on a hit with a melee attack. They also have the find familiar spell from the Ritual Caster feat (PHB, p. 169), choosing the wizard spell list. They were wondering, based around the wording of find familiar, if a touch spell from the familiar can still trigger the Touch of Death ability.

Can a touch spell delivered by the familiar trigger the Death Domain cleric's Touch of Death?

Yes (arguably), as long as the familiar is delivering a spell that involves a melee spell attack

The relevant section of find familiar says:

Finally, when you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell. Your familiar must be within 100 feet of you, and it must use its reaction to deliver the spell when you cast it. If the spell requires an attack roll, you use your attack modifier for the roll.

This line only applies to spells with a range of "Touch".

The Death Domain cleric's Channel Divinity option, Touch of Death, says (DMG, p. 97):

Starting at 2nd level, the cleric can use Channel Divinity to destroy another creature’s life force by touch.

When the cleric hits a creature with a melee attack, the cleric can use Channel Divinity to deal extra necrotic damage to the target. The damage equals 5 + twice his or her cleric level.

This only applies to melee attacks by the cleric. Since it's not specified, this applies to both melee weapon attacks and melee spell attacks.

One example of a touch-range cleric spell that involves a melee spell attack is inflict wounds:

Make a melee spell attack against a creature you can reach. On a hit, the target takes 3d10 necrotic damage.

Thus, the question simply comes down to whether it counts as the cleric hitting the target if their familiar is delivering the spell.

Since find familiar specifies that "If the spell requires an attack roll, you use your attack modifier for the roll", I'd argue that they're functionally equivalent. Thus, I'd rule as DM that any melee spell attack delivered by the familiar that hits qualifies for Touch of Death.

The counterargument

Of course, the reverse argument could be made, too. The familiar delivers the spell "as if it had cast the spell", so one could argue that aside from using your attack modifier, it's as if the familiar made the attack - and thus argue that if the familiar delivers a spell, the cleric is not the one making the attack so they can't use Touch of Death.

It's ambiguous, and could be interpreted either way. Thus, the DM and players should come to a ruling that makes the most sense to them, and be consistent with it.

No. Channel Divinity abilities are not spells

Forget the "touch" part - familiars can only cast spells, and Touch of Death is not a spell.

The find familiar spell description reads:

...when you cast a spell with a range of touch, your familiar can
deliver the spell as if it had cast the spell.

This feature of find familiar only works on "spells". A "spell" is not just any magical ability; it is a defined term. From the Player's Handbook, p. 202:

When a character casts any spell, the same basic rules are followed,
regardless of the character’s class or the spell’s effects.

Each spell description begins with a block of information, including
the spell’s name, level, school of magic, casting time, range,
components, and duration. The rest of a spell entry describes the
spell’s effect.

We know that Touch of Death is not a spell because it is missing some of the attributes described above: level, school of magic, and components. It also does not appear in the list of spells in the rule book.

Not a spell, therefor the familiar can't cast it.