To unlock 'Fishing in Baku' you must complete the first mission on Battlefield 4, on any difficulty.
It's a decent length mission, probably 25minutes on hard difficulty. However, don't be put off by hard, as hard is fairly straightforward.
With regards to weapon loadouts, I'd recomend the starting AR and grabbing a DMR from your first weapons chest, as it makes the headshots/scoring bonuses incredibly easy.
Just follow the mission to the end, and the achievement will unlock in the sequence between levels.