When Kennen, Keeper of Balance is played from Ava Achiever's ability, does Kennen's ability trigger twice, from being played and from attacking?

Yes — you get two separate instances of the ability, one for each condition met.

Kennen's "When you play me or I attack" is a single triggered ability but with two distinct trigger events, and it has no "once each turn" limit, so each fulfilled condition puts its own instance on the chain. The "when you play me" instance is placed when Kennen enters (Rule 383.4.a.2); because Ava plays him "here" into the battlefield where her combat is ongoing, he then gains the Attacker designation during the next Cleanup along with his controller (Rule 323.2.a), which places the attack-trigger instance (Rule 383.4.e.2). Arriving late to a combat still counts — that's the same principle as FAQ #4095/#716.

Each instance is a separate "you may pay [2] to stun a unit," so you decide independently for each: pay for one, both, or neither, and you choose the order if they're placed simultaneously (Rule 383.3.d). Note Ava plays him from hand, not from Hidden, so neither stun is restricted to targeting units at his battlefield. If he's played somewhere he never becomes an attacker (base, or a non-combat showdown), you get only the single play trigger.