How to avoid being decapitated by an executioner?

Solution 1:

There's no way to win against the Executioner - just evade his attacks for a time, and Alice will eventually get the chance to retreat. He's going to continue chasing you, so plan to continue evading his attacks and fleeing from him when you encounter him again.

The Alice Wikia notes:

The Executioner is completely invulnerable and the only means of keeping him at bay in order to move forward into more areas of the kingdom is to use his large scythe against his allies, as they are just as vulnerable to this attack as Alice.

Eventually you'll be able to defeat him, but I don't want to spoil his demise :)