Yes, you can branch off the existing supply lines by using a 'dishwasher' valve (we call it a 'stop') and it has two outlets stopped by one handle.
The plumbing drain can connect to the present drain pipes by using a 'continuous' waste. The new sink should come into the drain before the water goes through the P-trap. If you are using PVC (white) plastic drains you simply need to add a directional tee for the new sink to drain into. The directional tee has its branch entering the tee in a slight little curve which will drop the water into the P-trap rather than having it hit the backside of the tee and splashing. You can see a good example of the continuous waste under your kitchen sink.
You will find that cutting the hole to put the other sink in the top will be the harder job; but I'm betting you can do it.