When Aatchik enters, its first ability triggers, and Esix allows you to create a token copy of Aatchik for each artifact and/or creature card in your graveyard.
Put all but one copy of Aatchik into the graveyard due to the legend rule.
When each Aatchik token enters, its first ability triggers, and when each Aatchik dies, the last ability of each other Aatchik triggers.
Resolve all of the death trigger abilities, causing each opponent to lose 1 life each time and (for the surviving Aatchik's triggers only) putting a +1/+1 counter onto Aatchik.
If any opponents remain alive, resolve all of the enter trigger abilities, each time creating a 1/1 Insect creature token for each artifact and/or creature card in your graveyard.