Besides progression and special avatars the only way to obtain avatars is by finding them in Chests. Progression avatars are unlocked by reaching champion level ten with any champion.
Valentine's Day Avatars
These avatars were given out to everyone who logged into the game between February 14th 2017 and February 21st 2017.
These two avatars were reworked, see this reddit thread.
This avatar was originally meant to be the reward for players who played in Season 1.