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.
The Destruction of Lothering: As a non-mage, choose the Aggressive response to Wesley's threatening of Bethany, upon meeting Aveline for the first time for (+15). All other choices garner (+10).
Bait and Switch: Directly defend Bethany at the end of the quest for (+5)
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.
Portrait of the Past: Giving Bethany your mother's portrait (found in a chest in the Amell Estate basement) back at Gamlen's House results in (+15), regardless of dialogue choices.
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)
Act of Mercy: When talking to Grace, choosing "I won't help you" yields (+5) When talking to Sir Thrask at the end, turning the mages into the Circle yields (+10)
Long Way Home: When Merrill opens the barrier with blood magic, say "It did help us". (+5)
Bait and Switch: At the end of the quest, agree with Fenris by saying, "I know the risks" of cooperating with a mage. (+5)
Birthright: After finishing the quest, talking to Bethany and choosing "This will never be home" leads to (+10)
Enemies Among Us: When talking to Knight-Captain Cullen to end the quest, choosing "You're right" (+5) and "You can't take any chances" (+10)
Tranquility: After leaving Lirene's Imports, take an aggressive stance against the Fereldans (+5) Tranquility: When asked to kill Karl, choosing "Don't kill him" yields (+5)