I'm replacing an outlet under kitchen sink that garbage disposer and dish washer were plugged into because they stopped working. The old outlet was not GFCI. So I'm attempting to replace it with one. Our home was built in the '50s and has no ground line yet.
I used an extension cord to plug them into other outlets in the kitchen I knew had power and each appliance worked.
I opened the circuit breaker the outlet is connected to to remove power.
I then removed the wires from the old outlet but in the process of doing that the black wire that connected to the outlet the dishwasher plugged into broke and I could not tell whether it was from the line side or load side. There was no white wire connected to that outlet which I thought odd. I used a flash light to look in the box to find out where the black wire connected but did not find anything.
What I did see though is a set black and white wires looking like they were the line side that connected to the outlet the garbage disposer plugged into and a set of black and white wires that were capped off looking they were the load lines.
I connected the black and white wires from what I think is the line wire of the house to the top connectors of the new GFCI outlet as the diagram for it instructed. I then hooked the load wires to the other connectors as the GFCI diagram instructed.
I used electrical tape to tape off and screwed a plastic cap onto the copper ground wire.
I then flipped the breaker in the breaker box back to the on position.
I used a voltage tester to see 112v at the outlet for the dish washer but only a lightning bolt which I think indicates there was power for the garbage disposer. The switch for the garbage disposer was still in the off position. So I flipped it on and checked the voltage again and all I saw on the voltage tester is the lightning bolt.
I then decided to press the red reset button on the circuit breaker and when I did that the circuit breaker in the breaker box tripped off.
What did I do wrong?
Is the outlet the dish washer plugs into only supposed to have the black wire connected but no white wire?

What did you do wrong? Just about everything.

First, what you describe is installing a GFI outlet, not a GFI breaker. A breaker would be installed in your breaker panel.

I think you are incorrectly using the terms line and load. There is no line or load on a regular outlet. There is however, a line and load is on a GFI outlet. That's OK, but I can't figure out if you really mean hot and neutral.

The outlet you are replacing has been split. That is to say that the terminals on the hot side have been separated and half of the outlet is use in conjunction with the switch for the disposal and the other half is always hot, and used by the dishwasher. You cannot do that with a GFI outlet.

In the situation you described there has to be at least two white wires.
One of the white wires would be the neutral and would be connected to the neutral side of the outlet (the side with the silver screws). The other white wire would be used to take power to the switch and would be connected to the black hot wire coming to the box. The black wire coming back from the switch would be connected to the hot side of the outlet (the side with brass screws). The neutral white wire coming from the power cable serves both half's of the outlet. Take note that the tab that connects the two hot screws has been removed. Power goes directly to one half of the outlet. Power to the other half has to go through switch first.
This is called splitting the outlet. You cannot do this with a GFI outlet. The bare wire is the ground wire and needs to be connected to the outlet under the green screw. It is a safety device but has nothing to do with the function of the outlet.

You need to determine which cable goes to the switch and which is power coming to the box. Then tell us what wires you have in the box and we will help you reinstall the old outlet.

If you want these to be GFI protected you will have to install a GFI breaker in the breaker panel. Personally, I wouldn't worry about it. These are not things that you normally plug and unplug nor do you normally come in contact with the electrical components.

EDIT: When you say you press the reset on the breaker, if you really mean on the breaker, you are already GFI protected.

I did not proof read my posting enough to catch my errors. When I said GFCI breaker, I meant outlet.
I did some more research and found that GFCI outlets are not recommended for use with dishwashers and garbage disposers. So I took it out and put the old outlet back in and wired it up as it was with the exception of the ground wire because it broke off it looks like outside the metal wire box. I need to re-read your answer again to look more closely at what you said to make sure I understand it.

If you have everything reconnected all is good. The ground wire should have also been attached to the metal box. If you can get to it you can add to it with another piece and a wire nut