Step 1: Find Suitable Bottle
I used an old perfume bottle, but pretty much anything will work. I found sample bottles from hotels to be especially useful for the small potions.
Step 2: Make/Modify Cap
I was lucky enough to find a bottle which already had a large cap. I simply painted it brown and tried to make bubbles in the paint for the circular "cork" effect. I presume it would not be terribly difficult to carve an actual cork to fit your bottle, if you so choose.
Step 3: Color and Fill With "potion"
Just mix water and food coloring to whatever hue you want. The bottle I used had a very small opening, being a perfume bottle, so I had to use a pipet to get it inside.
Step 4: Label and Decorate
To get that worn look for the paper, fill a large baking pan (one with a high rim) with coffee. Repeatedly soak a sheet of printer paper in it, and let it dry, until the paper is the desired darkness. I found a heat light facilitated the process, but one is not necessary.
I added the string as a finishing touch, it is just rough gardening string, but I think it brings the whole thing together, as well as cover up the rim of the bottle under the cap.
Step 5: Display
You obviously don't have to keep it in the snow, but I liked the snowy Skyrim feel