Greatness combines multiple measures into a single-number that summarizes your PvE prowess. It is calibrated so that anything above 1.0 is really good.
While, like the PvP measure, it puts some emphasis on your ability to kill stuff, it puts extra emphasis on your ability to finish difficult tasks (nightfalls and raids) and work with teammates (assists, drop orbs, and resurrections).
The general formula is: Greatness = Basic + Nightfall + Raid.
0.2*kd + 0.15*spree + 0.15*precision_kills + 0.15*assists +
0.1*resurrections + 0.1*orbs - 0.1*suicides
This is weighted by how many strikes and story missions you have done (your experience): if you have done more than 200, you get a weighting factor of 1.
Nightfalls and Raids
These are solely counted by number of clears, with up to 0.15 contribution from raids if you have done more than 5, and up to 0.1 if you have done more than 10 nightfalls.