Support XP comes from building, not from rifle kills. Every hammer swing on an unfinished piece pays 1 XP, a swing on the yellow X marker pays 7 XP, and completing a piece pays 15 XP. Pallet and crate deliveries, vehicle repairs and kills with shotguns, LMGs, launchers, mortars and anti-air also count.
The fastest route is therefore cheap, repeatable pieces near active fighting: sandbags and barbed wire cost 10 and 5 supplies, need no FOB and can be chained from a backpack of loose supplies. Support 8 sits around 25,000 XP; players report reaching it within a couple of long matches of dedicated building.
Until you get there, you do not need a Large hammer to help build heavy pieces. Any hammer can finish a blueprint that a Support 8 teammate placed, so the usual split is one placer and several finishers.
Sources: s1-changelog · steam-skivin · allthings-cluster