Lets see the checklist...

[Siber Fox]

How does one get the check list (it appears as a green check when you complete it) in the mission objective?

[MadFred]

well first you need to have the objectives in your briefing,

then you can check them by putting this in a trigger or waypoint or whatever when the objective is achieved:

"1" ObjStatus "DONE";

where "1" in this case is objective One, obviously.

[macguba]

Read snYpir's tutorial "how to use objectives and init.sqs"

There is a whole sub-category on Briefings and Overviews in the Ed Depot tutorials.