only when you have a high level friendship you'll get their picture in the mail when they move away. though I did not have a high friendship with a certain villager that moved away but ended up getting his picture in the mail anyway but that was probably because I gave him medicine when he was sick which is another way to obtain a villager picture.
They don't always give you a picture, but if your friendship level is high enough when they move out they'll send one in the post with their goodbye letter.
There's also a way to tell with the megaphone. If you have them on-screen and call their name (don't be so close that they get startled and mad), they'll wave and cheer if you have good friendship. If you don't, they'll just look around.
The coming over to your house uninvited is a better way to tell, but that happens randomly kinda.
the megaphone is the best way to tell immediately, i've heard. if you want easy friendship points, whenever they ask for furniture or fruit, just refurbish or give them a perfect fruit.