It would work, once. The Lich would come back to unlife.
The Rejuvenation trait of the Lich says the following:
Rejuvenation: If it has a phylactery, a destroyed lich gains a new body in 1d10 days, regaining all its Hit Points and becoming active again. The new body appears within 5 feet of the phylactery.
The destruction of a lich’s phylactery otherwise has no effect on the lich, aside from the eventual degeneration into a demilich if they don’t construct a new one.
The Clone spell says the following:
At any time after the clone matures, if the original creature dies, its soul transfers to the clone, provided that the soul is free and willing to return. The clone is physically identical to the original and has the same personality, memories, and Abilities, but none of the original's Equipment.
As a result, if a lich who has cast the Clone spell has its phylactery destroyed and is then killed, its clone will come to life, and will possess the same stat block it did before. As a result, it will still be a lich - the clone that animates would remain undead.
Note, however, that a lich can only ever do this once, since it will no longer be a living creature after its transformation into a lich, and it thus cannot benefit from a second casting of the Clone spell.