You can chop all of the vegetables finely by hand or alternatively chop them in a food processor.
If using a food processor cut everything into small chunks and only put in a few handfuls at a time, pulse until you get a consistent small size. If you process everything at once the colors will start to run together, I recommend doing the cabbage, radish and carrots separate, then the bell pepper and chili together and lastly the cilantro, green onions and red onion.
At this point the salad can be chilled until you are ready to eat, dress at the last minute so the water is not leached out of the vegetables.
When you are ready to eat, squeeze the limes straight into the bowl of vegetables along with the olive oil and salt, stir everything together and serve with tortilla chips.