Do challenge runs count towards Isaac’s completion achievements

No, you cannot. Sadly, the achievement only registers on the boss kills, and not by going further down the dungeon.

All kills except for flies (both the harmless and aggressive flies) count towards charm of the vampire kills.

This includes splitting apart enemies that can be split (like The Fallen or Envy, but not the twin flies).

Each entity will count as one kill when depleted to 0 health. This means that for example, a Mr. Maw (being the fact that it's actually two entities combined into one enemy) will count as two kills when you fully kill it (even if you kill it while it's head is on its body). This also means that killing an enemy that splits into two upon death (like The Fallen) will count as 3 kills, one for the first enemy, and two for the second two.