The way I go up a kerb is to slow on the approach and keep the crank arms horizontal (I keep the right pedal forward but that's just preference), then step on the pedal and pull the handlebars up to lift the front wheel over the kerb. Then as the rear wheel approaches just move your weight forward and give the pedals a push to keep the forward momentum. Round my way they aren't 6", so make sure you've got clearance so you don't catch your chainrings (as someone already said) and be prepared to jump off the saddle in a hurry if you're not familiar with how the bike will react. If you can, practise on smaller steps.

Going down is easier, again I slow on approach (probably not necessary for lighter riders), shift my weight back as the front wheel drops down, then shift my weight forward as the rear wheel drops down. Here the idea is to minimise the impact as the wheels go over the step.

Chances are my bike would cope with treating it much more roughly than this, but the rider might not.