Why do black lines against the sky show a purple tint in my photo?

This is known as purple fringing. It is occurs because you have a region of very high contrast. It is normal that this occurs in extreme cases but better lenses show less of it.

The good news is that software like Lightroom have tools to deal with this automatically. The Lightroom tool in particular handles not just purple fringes but other colors due to a more general problem called Chromatic Aberrations.

Some digital cameras, mostly from Panasonic and Fuji, handle it completely internally without user intervention or the option to disable it. On other models you can enable in-camera correction such as Lateral Chromatic Aberration removal but it slows down the camera significantly and reduces the buffer-depth.

Wikipedia explains it well and even has some suggestions to minimize it, like avoiding over-exposed areas and avoiding to shoot wide-open (wide apertures tend to show this problem more).