Half a dozen undead creatures in Volo’s Guide to Monsters and Mordenkainen’s Tome of Foes receive significant updates in Monsters of the Multiverse, and deathlocks account for half of these—unsurprisingly, since they’re all spellcasters.
Gone from the deathlock’s Spellcasting repertoire are arms of Hadar, hold person and chill touch. Eldritch blast is reskinned as the ranged spell attack Grave Bolt, dealing an extra 3 damage (presumably from the deathlock’s Charisma modifier). A new Multiattack lets it attack twice with either Deathly Claw or Grave Bolt, doubling the amount of damage it can deal in a single turn.
These changes turn the deathlock inside out. For starters, it loses both of the spells that benefited from being boosted to a higher level by the deathlock’s warlockitude. It also no longer has anything that fills the role of chill touch’s suppression of healing. On the other hand, the fact that the deathlock now gets to attack a second time makes invisibility-based ambush more practical (although it gains advantage only on the first attack roll of the two), and spider climb no longer has to compete against more potent spells for the use of a spell slot.
As for direct attacks, the choice is no longer between Deathly Claw and chill touch but rather between Deathly Claw and Grave Bolt—which is really a choice between melee and ranged combat. This choice is resolved by looking at the deathlock’s ability scores and asking what they say about its combat role. With Charisma as its primary offensive ability and Dexterity as its primary defensive ability, the deathlock is a spellslinger, and as such, it wants to sling spells and avoid melee.
Therefore, its strategy is now to fortify itself in advance with mage armor and either disguise self or invisibility (the latter precludes the use of detect magic while the deathlock concentrates on it); stay as far as possible from likely foes; cast hunger of Hadar to delay opponents while the deathlock completes its task(s); and if that fails, cast spider climb to escape or to attack with Grave Bolt from inaccessible places. Since the deathlock no longer has a convenient way to paralyze an opponent, Deathly Claw is now only a last-ditch defense, for use when the deathlock is cornered and can’t get out of melee.