just some things that have worked for me..
when they ask for any fruit, give them foreign perfect fruit.
when they ask for furniture, give them customized items OR an item from their desired birthday gifts list, located
here.
run errands for them, play hide and seek, and let them visit your house, etc. if you can't go to their house at whatever time, be up front and say no instead of saying yes and then not showing. that lowers friendship, while saying you're busy does not.
buying their items or letting them buy stuff off you DOES NOT, in my experience, increase friendship. and saying no does not lower it.
when you write them letters, it doesn't have to be an extremely long letter. my usual letter consists of "best friend", "friends", "present", "*villagers name*", "thank you" all on different lines. this always works for me and they mail me nice presents back. you can also attach a piece of furniture, but not attaching one also increases friendship.
to check friendship levels you can..
stand near them (not TOO close, but so you can see them in the frame) and call out their name with the megaphone. if they bow/wave to you, you have good/high friendship with them.
be in your house. if you have a maxed or near maxed friendship level with a villager, they will come over uninvited. (i.e., HEY *mayor name*!!! COME OUT AND PLAY!")
that's about all i can think of at the moment. i hope this helps someone. good luck with pics! <3