Water filters are only designed to filter water; the bad elements of normal tap water are present in concentrations of less than 1 part per 100,000 and are usually more in the range of 1 part per 1,000,000. A water filter is usually very efficient at removing organics and chlorine but they don't have much capacity so a really concentrated solution is going to pass through the filter without getting cleaned.
It is possible to remove most of the colour from water using a bleach filter, but household water filters cannot remove the flavour from flavoured liquids. Vaporization is the best method to complete that process.
If you have the time, and inclination, you could use a slow sand filter. Slow sand filters are usually used in water purification for treating raw water to produce a potable product, but you could try it on the liquid you are experimenting with.
Alternatively you can distil the liquid and remove water from the mixture to get clean water and leave a residue mixture behind. This process filters out particles and the colour will fade. This is because the fluid will lose some of its solid state altering the taste of the liquid.
To filter out all of the colour is probably not possible, but you can try to separate the colours by using several chromatography techniques. The most common is paper chromatography is an analytical chemistry technique for separating and identifying mixtures that are or can be coloured, especially pigments. This can also be used in secondary or primary colours in ink experiments. If all you’re not interested in the fluid you can simply boil the liquid until all that’s left is the residue of the colouring and flavour.