procedural-planet | Proceural planet in WebGL and three.js | Graphics library

tldr: I would use a perlin noise generation with some tacked on biomes.

This article/tutorial goes over code snippets and their implementation methods. Suggesting the best algorithm for your task depends entirely on your skill and end result goals.

However a brief description of perlin noise and using it with realistic aims in mind...

As with most terrain generation, noise functions are your friend - Perlin and/or simplex noise in particular. I've implemented some planetary terrain generation algorithms and although they are in 2d, the resulting height / "texture" map could be projected to a sphere rather easily. I assume conversion to hex format is not an issue either.

My technique has been creating multiple noise layers, e.g. temperature and humidity. Temperature is fused with a latitude coordinate, in order to make the equator more hot and poles cold, while the noise makes sure it's not a simple gradient. The final terrain type is selected by rules like "if hot and not humid then pick desert". You can see my JavaScript implementation of this here: https://github.com/tapio/infiniverse/blob/master/js/universe/planet-aerial.js

As for the water percentage, you can just adjust the water level height as noise functions tend to have a constant average. Another option is to apply an exponent filter (useful also when generating clouds, see my implementation here).

Another way to generate spherical terrain that came into mind (haven't tested) is to use 3d noise and sample it from a surface of a sphere, using the resulting value as the ground height at that point. You can then weight that according to amount of water on planet and the latitude coordinate.

I'll end with a link to one practical implementation of 3d planetary terrain generation: http://libnoise.sourceforge.net/tutorials/tutorial8.html

To generate any random style of realistic terrain you are going to have to use noise of some kind. In past projects I myself have used the diamond square algorithm. However that was to simply generate heightmaps.

For some more light reading I would check out this article about realistic terrain techniques.