It's very simple:
If the bowl is unvarnished, you can probably start colouring straight away. Otherwise roughen the surface slightly with a sheet of fine sandpaper: then the colours adhere very well.
The design shown here is very simple but very effective: gentle waves in various colours. The lines should touch here and there; then fill in the areas between them with colour, but always allowing the wood to show through in places. You can practice beforehand on paper, until the width and proportions look harmonious.
When you’ve finished colouring, dip the brush in water and gently work over the lighter colours until they flow. You get a very attractive watercolour effect, and with a bit of practice you can control the extent to which the colours mingle.
Finally we would recommend varnishing the bowl with an environment-friendly and non-toxic varnish – unless you intend using it purely for decorative purposes and it’s unlikely to come in contact with moisture. Incidentally, the pigments used in the pencil leads are quite harmless.
Be brave: experiment! You can modify the design as you like, by making the waves look stronger, using different colour combinations, or arranging the pattern differently.