Bow names: Paris, Paris Mark one, Cernos, (others exist, please add in comments)
First, you'll need to buy a bow, unless you picked one up in the first mission. The Paris Mark one is available for 15,000 credits. You could also buy the blueprint for the Paris and assemble it. The materials for it can easily be found on Mercury and Venus. The Paris has better stats, but the Paris Mark one is fine. If you have a high rank or are willing to spend platinum, other bows are available.
Second, Equip and use the bow. Weapons gain small amounts of XP for reaching the end of a mission, and more for reaching the end of a mission after making kills/inflicting damage. The bow's first levels go quickly, but the later ones can be a big grind.
(Edit: Partly patched)If you want to
shortcut your way into unlocking
Have your bow reach lvl 30 in the end of mission rewards.
If your bow reaches level 30 during a mission, go to dashboard and kill warframe immediately. Repeat the mission, and your bow will count as a second one reaching level 30