Feathers only appear on the Endless Slope. Each stage contains 3 feathers so there are 15 total available from stages 1-5 and then I believe there are another 3 in the endless nighttime section that follows stage 5. I'm pretty sure this is correct but please let me know if I'm not.
When you collect a feather a pop up will appear in the top left corner showing which of them you've got from the current stage. If you get all 3 then it will activate a bonus which gives you some extra leaves at the end of the stage.
Feathers can show up in numerous positions. You will need to jump to reach them quite a lot of the time. They are commonly tucked in close to obstacles like rocks, tree stumps and the posts of the bridges that appear from stage 3 onwards. They can also float in from either side of the screen or down from the top. They often appear at the bottom of the little drop with hoop bits which can be frustrating because if you've already jumped for the hoop then you may be off target for the feather. They can also appear underneath branches where you need to duck. Occasionally these branches come with a choice to jump or duck them and the feathers are in the jump sections.
Most of the time they can be spotted from a distance giving you time to move into position but can appear quite quickly and be difficult to see. Vision is especially bad on stage 3 where the gold colour of the feathers can easily blend in with the trees on the sides of the slope.
I would recommend using the astronaut character on runs where you're going for feathers because his double jump ability makes it easier to move across the screen quickly and get out of the way of obstacles. This is especially helpful on stages 4 and 5 where feathers can appear above the chasm sections and you'll need to jump quickly to reach them.
If you're trying to get 15 feathers then you'll want to get as many as possible from stages 1-5 and then mop up the couple you miss during the nighttime section. If you miss more than 3 from stages 1-5 then you might as well restart the run. Again if I'm wrong about this and there are actually more than 18 total feather available then please let me know.
The achievement will pop when your run has finished.