There's a thread full of good information and techniques to try out here:
http://pixelation.org/index.php?topic=19594.0

Generalized advice is always hard to give, we need a direction to critique you first! I reccomend starting with some anatomy studies though.

Honestly you can use anything for making tilesets.
Aseprite has a few features specifically aimed at that, but all it really requires is setting up a defined measurement (for example, 64x64 pixels, my standard) then sticking to that while creating the individual tiles.

If you're looking for SPECIFIC tools though, then aseprite is probably a good bet, and honestly I reccomend it to most people anyways.

think about the mechanics of breathing, and study other things that have already done it!
Mechanically, it's an expansion of the upper chest and a tensing of the gut, as seen below:


And other mediums that have used breathing motions for more interesting idle animations are numerous! For example, here's Balrog, from street fighter:

Here, instead of emphasising chest expansion and the breathing itself, they instead used it to create a posture change that gave him a more guarded, trained look. They minimized it in favor of a boxing bob!

And here's blanka, with emphasis on his chest and shoulder movements to create a more bestial, almost gorrilla-esque appearance.
Think about the personality you want to portray, is she a powerful woman? Meek and unassuming? Calculating? Try and represent that through the way she breathes and moves!

That's a pretty chunky reindeer! Find a reference image off google, deer are very slender creatures!

It's a bit strange here, actually. I think the dithering was what pushed that somewhat painterly vibe as now i get more of a cartoonish celshading vibe off it.