How can I earn achievements I completed while achievements were disabled?

That was a server fault, not their decision.

While achievements were disabled, you did not earn any. (Plus "Kerrigan Power 70" may be broken on its own, as in "only awarded when you get it after the last level-awarding mission, not from archives".)

• All the level-specific achievements you can obtain by replaying missions from archives.
• Mutation achievement ticks when you swap a mutation for another one.
• For some achievements, like conversations, you will need to restart the campaign. "Abathur's Cookbook" also seems to be awarded at evolution choices themselves, which are not replayable.

As for now, Blizzard did not announce anything regarding fixing the achievements in HotS, so we have to assume it was a usual "just do it again later" server outage.

At least you can do Kerrigan Level 70 on Casual :)