So what I use:
A good nail polish remover. Lately I have been using Depend´s Super Quick Nail Polish Remover.
A sheet of paper tissue or paper towel
Something for nourishing my hands and cuticles after the cleaning process.
1. I pour out some nail polish remover. I use the cap of the nail polish remover bottle.
2. For cleaning up the bigger mess I dip q-tip into the nail polish remover and swipe away bigger mess.
3. For precise cleaning I take the brush, dip it into the nail polish remover and carefully clean my cuticle area. I the paper towel to swipe the brush clean.
4. After I have finished cleaning I use cuticle oil and/or hand cream to moisturise and nourish my hands and cuticle area.
And here is video about how I do it: