The first problem is that you may be using the wrong cable and breakers. NEC calls for 2 20Amp small appliance circuits in the kitchen, to accomplish this you'll need to switch to 20Amp breakers and #12 wire.
The next problem. You'll have to pull new wire anyway, if you want to hook up GFCI receptacles. GFCI receptacles will not work properly with a shared neutral, you'll end up with nuisance tripping with a shared neutral. GFCI receptacles work by monitoring the balance between hot and neutral, so if the neutral is shared the GFCI will not work properly.
To wire up the kitchen properly, you'll have to pull 2 new 12/2 cables from the breaker to the kitchen (all #14 wire on that circuit will have to be replaced). Then install 2 20Amp breakers, to supply the kitchen. You'll install the GFCI's as the first receptacle on each circuit, which will protect all downstream receptacles.
You can share a neutral between 2 GFCI receptacles. The catch is you'll have to pigtail the neutral to the receptacles, not use the neutral from the load side of the first GFCI to feed the second.
So you should be able to do something like this...
![enter image description here](https://i.stack.imgur.com/nKHGX.png)
But not like this...
![enter image description here](https://i.stack.imgur.com/2MmjH.png)
You'll then be able to use the load side of each receptacle to feed other devices, like this.
![enter image description here](https://i.stack.imgur.com/567iH.png)
I had a similar situation in my bathroom. I had power in the ceiling, to the light, and a 14/2 running down to the switchbox. My assumption is that the original install consisted only of the light switch, but had been "Upgraded" by everyone's favourite contractor, Some Moron. This esteemed craftsman put in a combo switch/receptacle, Jumpered the hot side of the switch to the hot side of the receptacle and then pigtailed the neutral of the receptacle to ground.
IF THIS IS YOUR SITUATION,
You need to run a new 14/3 from the ceiling to my switch location and break drywall to install a 2 gang box. The final wiring is:
Panel Black - 14/3 black. (with pigtail in switch location)
Light Fixture Black to 14/3 Red. (Switched Hot - power to light)
Panel White - 14/3 white.
In the 2 gang box, I took a black from the pigtail to the GFCI hot, and the white from the 14/3 to the neutral screw.
Other possibilities:
Existing 14/3 -- Do as above, but obviously you don't need to run a new cable. Check the wiring in the light box to make sure you get the switched hot right.
Double 14/2 - In this situation, you'd have 1 14/2 powering the receptacle, and another 14/2 acting as the runner for the switch. Usually black is hot, and white is switched hot. The white wire should be marked as such, perhaps with a piece of tape on the end. Wire it exactly as it is.
Power to the switch box
You may have the situation where the power feed from the panel goes directly to the switch box. Usually you'll have a feed from the bottom (power) and another pair going out the top to the light. Check these with a voltage tester!
Create a pigtail on the feed black with two short pieces of black wire. One goes to the switch, the other to the receptacle. To the other side of the receptacle, attach the black to the light. This is your switched hot.
The white from the light, the white from the feed are wired together with an additional short piece of white wire which goes to the neutral of the receptacle.
Additional Warnings:
All connections go to the LINE side of the GFCI.
Do not work with power on. Turn off the breaker or fuse.
If you don't see one of the scenarios presented here, then call for professional help.
Best Answer
Both the black wires currently on the receptacle, need to be connected together with a pigtail. The pigtail will then connect to the brass colored LINE terminal on the new GFCI.
The white wire from the old receptacle, will attach to the silver colored LINE terminal of the new GFCI.
The grounding conductors should also be pigtailed, and a pigtail terminated at each devices grounding terminal.
The wiring seems a bit strange in that box, but I'll have to think about it a bit more before I decide if it's dangerous.
In the meantime, you should use a black marker or tape, to mark both ends of the white wire that's attached to the switch.