I have a ceiling fixture in middle of kitchen. Would like to install light over sink. Can I use the wiring in the existing light to wire the light over the sink and have a separate switch for each light? I don't want both lights to come on when I flip the switch for the ceiling light.
Assuming the Fixtures are all on a 15AMP Circuit and using 14/2 wire.
The Complexity would depend on how the Switch was wired. Power to the Switch, Switch Leg to the Fixture, or Power to the Fixture and Switch leg down and back. If you have the Power at the Fixture then you won't have a Neutral at the Switch. If its a newer house you should have a White wire in the Switch box that is just passing though and only connected to other white wires.
If you have a Neutral in the Switch box, and If you have Access to Drop a second 14/2 from the new Fixture to the existing switch then sure. You just Connect up all the Neutrals, then get a Switch like this
Connect the Existing Hot Switch leg to one switch and your new Fixture's hot switch leg to the other switch.
If there is no Neutral then you have two options:
Option #1 Branch the Incoming Power from the Existing Fixture down using a 14/2 to the Existing Switch. Then You will have a new HOT and Neutral in the Box. Then add a second 14/2 up to the new Fixture. Use the same Switch as above though remove the interlock between them so you can feed separate HOT feeds.
Option #2 is to do as they did and branch from existing Fixture to new Fixture. Take the incoming hot at the old fixture, Tap off it to the new fixture. You will need a 14/2 down to the switch in this case too. You connect the HOT from the existing Fixture to the HOT of the line you ran to the switch box. Then use the same switch though you are going to remove the interlock between them so you have an incoming hot for each fixture. Then use the white wire after its labeled to provide power back up to the Fixture.
This is what the OP's existing kitchen light wiring looks like:
Switching on the Light bridges C to 1 (C to 2 doesn't do anything because 2 doesn't lead anywhere).
Adding a second Light and Switch:
Each Switch controls its own Light independently from one-another.
If you instead only want 1 Switch but 2 Lights, take the red wire from L to L and connect it into A to A.
Sorry for the necro, but I thought my answer could help future page visitors.
You can't use the wiring of the existing light. Rather, you have to tap into the junction box that the lighting fixture connects to. However, that will only get you the second light on the same circuit. As @DoxyLover points out, if they're on the same circuit, you can't use separate switches for each light.
You could use one switch for the main circuit, then a second switch just for the sink light. But that means the existing light would have to on be when the sink light is on.
However, if you can instead connect to a junction box housing an outlet nearby, you could find a light fixture with a wireless switch. Here's one product from Amazon: http://www.amazon.com/Basic-Wireless-Light-Switch-Kit/dp/B005B0AY0K
Alternatively, you could use a Wi-Fi enabled light bulb that you could turn on/off with your phone: http://www.amazon.com/LIFX-Edison-Enabled-Multicolor-Dimmable/dp/B00EJ9FY2C
The reason the latter two options will work is that the outlet junction box is not (likely) on a switched circuit, so the fixture will always have access to power--unlike the existing light circuit that, if the switch is off, will not get any power.