What the achievement asks
Collect one million Azerite on Island Expeditions. Ten points, account-wide.
The number sounds enormous and is not. Azerite collected on an island is the score you race the rival team for, and an expedition is won at roughly twelve thousand, so a million is on the order of eighty completed runs. It accumulates across every expedition, at every difficulty, forever.
There is no way to hurry it
This is a pure counter. Azerite comes from killing things and gathering what they drop, which is what you do on an island regardless, so there is no technique to apply and nothing to seek out.
What does change the rate is finishing rather than leaving. A run abandoned halfway still counts what you gathered, but a run taken to the end banks the full score, and the last stretch is the fastest part because both teams are killing at their peak.
Do it alongside everything else
Eighty-odd expeditions is close to what Expert Expeditioner already asks for, so if you are chasing that one, this arrives on its own before the fiftieth Mythic win is far behind you.
The same runs feed Metal Detector, Give Me The Energy and the eleven per-island achievements. Nothing in this category rewards farming it in isolation.