Fighters, Brawlers, Gunslingers, Slayers, Avenger Vigilantes, and Oracles of Battle can get Improved Critical at 8th level.
Fighters, Brawlers, and Gunslingers gain a bonus combat feat at 8th level. This assumes you haven't selected an archetype that loses their 8th level bonus feat. Since they have a BAB of +8, they qualify for Improved Critical, provided that they select a weapon for which they are proficient.
Prerequisite: Proficient with weapon, base attack bonus +8.
Slayers gain a Slayer Talent at every even-numbered level. If select Rogue Talent at 8th level, they can use it to gain the Combat Trick rogue talent, which can then be used to gain Improved Critical because they have a BAB of +8.
Similarly, Vigilantes choose a Vigilante Talent at every even-numbered level. If they chose the Avenger option for their Vigilante Specialization, then they will have a BAB of +8 at level 8, and can choose the Combat Skill talent to gain Improved Critical.
Also, multiclassing any of the above classes with other full-BAB classes, depending on the build, can get you Improved Critical at character level 8. If the class you choose at 8th level gives a bonus feat option, then you can select Improved Critical. Examples:
- Paladin 6 / Fighter 2 (choosing Fighter for your 8th character level)
- Cavalier 4 / Slayer 4 (choosing Slayer for your 8th character level)
- Barbarian 3 / Brawler 5 (choosing Brawler for your 8th character level)
An Oracle with the Battle mystery has access to the Weapon Mastery revelation:
Weapon Mastery (Ex): Select one weapon with which you are proficient. You gain Weapon Focus with that weapon. At 8th level, you gain Improved Critical with that weapon. At 12th level, you gain Greater Weapon Focus with that weapon. You do not need to meet the prerequisites to receive these feats.
This revelation may be taken at any level, including those below 8th. As soon as the Oracle reaches 8th level, the revelation grants them the bonus feat.
Honorable mentions:
If you use the retraining rules, then any character with a BAB of +8 can pay a cost to exchange one of their feats for Improved Critical.
Swashbucklers get a 5th level class feature that functions as Improved Critical with one-handed or light piercing melee weapons, but only while wielding those weapons.