It seems to me the simplest way would simply to have a thinner atmosphere on the planet.
If you look at a blue sky it shades from pale blue near the horizon to dark blue near the zenith because of the different atmospheric distances that light has to travel. The farther light travels through the atmosphere, the more it is scattered and the lighter the sky in that direction looks.
Making the atmosphere a bit thinner but still stick enough to breath should make the sky a darker blue and thus closer to purple.
Then you would need to add a lot of red light so that the sky turns from deep blue to purple. Suppose that a lot of land plants and water plants are red. Then the sun shining on the plants would be reflected up to the clouds and down from the clouds as reddish light, mingling with the dark blue of the sky to look purplish.
Sunrises and sunsets are often pink and purple. So maybe the habitable zones of your planet have perpetual sunrise/sunset conditions and perpetual purple skies.
If your planet orbits a dim red dwarf spectral class M star, in order to be warm enough to be habitable it will probably have to orbit so close to the star that tidal forces will tidally lock the planet. slowing the planet's rotation rate so that a planetary day equals a planetary year. Considering how close the planet would have to be to the red dwarf star and how short it's year would have to be, such a slow down would not be as much as slowing Earth's day to a year.
Once the planet's rotation rate is slowed so that a day equals a year, one side of the planet will be in perpetual daylight and heat up greatly, and one side of the planet will face away from the star in perpetual night, and will cool off.
It is possible the air and water from the hot side will all flow to the cold side and freeze solid, leaving the planet airless and uninhabitable. But possibly the atmosphere and hydrosphere will transfer enough heat from the hot side to the cold side to keep the atmosphere and water from freezing on the cold side. In that case the twilight zone between the hot side and the cold side may be the only part of the planet with the right temperature for life.
In the twilight zone the red star will be at the horizon. The red star's light will contain many different frequencies of visible light, and should have enough blue light for atmospheric scattering to make the sky dark blue. But the vast amount of red light from the red star should also cause a lot of red light to be scattered by the sky, perhaps making the sky appear purplish.