The datasheet is quite large but it basically seems to be a case of initialising things, turning the sequencing engines off if not used, then just poking PWM values to set LED brightness.
Unless someone has already done it with a PICAXE it is probably a case of going through what example code you can find working alongside the datasheet to figure out what initialisation values should be.
If you are planning to use the LP55231 for a project you will probably have to do that anyway to get an understanding of how it works to get things working how you want. If you do just want to turn LED's on and off it might be better to choose something else, something which is better understood in the PICAXE community.