The rules state that a figure can perform two actions on its turn (in skirmish), but can neither attack twice nor perform the same special action twice. Can a figure use one action to attack, then a special action which allows more attacking? For example, could Luke attack and use the Saber Strike special action in the same turn, or could Vader attack and use Brutality?

Nope. This bit from the skirmish rule book specifically prohibits that:

Attacking: Figures can only use one of their actions to attack
per activation. This includes using special actions that involve
performing one or more attacks (such as the Nexu’s “Pounce”
or Darth Vader’s “Brutality”).

Got it - thanks!

However some actions that cause damage are NOT attacks - like Chewbacca's slam and Vader's force choke. Vader can do Brutality and then Force Choke, so could kill 3 different characters, potentially.