Friendship with Bethany can generally be earned through acts of kindness, by siding with mages or by supporting family. Bethany is a faithful Andrastian and also distinctly opposed to blood magic, so rebuffing demons and opposing blood mages will earn friendship as well.
Birthright: By simply entering the vault in the Amell Estate basement you gain (+15). A further (+5) can be gained upon returning to Gamlen's House with Grandfather's Will and speaking with Bethany by choosing the friendly response on the second dialogue wheel.
Tranquility: Promise Lirene you won't harm Anders (+5) Tranquility: Upon leaving Lirene's Fereldan Imports you will be approached by Angry Fereldan; choosing the response, "We're on the same side." (available to a DiplomaticHawke) instead of deferring to Bethany results in (+5). Alternatively, deferring to Bethany will result in the same approval gain, though Aveline Vallen does not gain (+5) also. Tranquility: When talking to Anders, choosing "You want me to fight templars?" yields (+5) Tranquility: When asked to kill Karl, choosing "Do as he asks" yields (+5)