Although Aaronut is entirely right in stating that you would be better off using alternate colourings that do not damage the texture of your icing, you can use normal food colourings, I do quite often.
When I use them on fondant or marzipan, the main issue is the capacity of the icing to absorb the colouring. They have limited ability to tolerate the liquid, so if you need a deep colour, expect a bit of a fight.
The best method I have come up with, is to treat it like pulling candy. Put the required amount of icing in a bowl and make a well and enough material to fold over that well. Put in a small amount of colouring and fold over. I tend to start off by squidging it about in folding motions to get the liquid in to the material, it'll be quite tacky at first. Then I roll it between my hands in to a sausage, fold in half lengthways, roll, fold, roll, fold...
It'll marble up to start with, given the folds, but eventually become one solid colour and will become less tacky with each fold, in a similar way that kneading bread makes it progressively more coherent.
Once settled, roll towards a ball/block rather than a sausage, then you can roll it out ready for use. If you are a very minute amount over the moisture level that you can roll it, try to incorporate a small amount of icing sugar (confectioner's) whilst folding to dry it out, like adding flour to bread.
It works, but is messy, take all rings off and expect to need to wash your hands a lot.
Edit
Found a video that roughly shows how I would do it, except the demonstrator has a different way of applying the colouring to the fondant initially and is using a much larger quantity than I tend to, so does not hand roll in the air as much as I would or as fast.
http://www.ehow.com/video_2333485_coloring-cake-fondant.html
Meringue is a protein-based foam. The proteins form a semi-stable mesh with air bubbles trapped inside. The reason they are semi-stable is that they unravel a bit and hook into each other, Velcro-style. When you introduce fat, it lubricates and smooths the unraveled proteins, so they lose their hooking ability. If the fat is introduced before the eggwhites are beaten, there is a fair chance that you won't get a foam at all.
It is possible to mix the prepared foam with fat without destroying it, although it will lose some volume. This is usually done when the bulk of the mix is not the meringue, but some kind of creamy filling which gets lightened by the air added through the meringue. These fillings are usually set through baking (such as a cheesecake filling), or gelling agents (there are cremes which combine egg whites foam, fat-containing dairy products, and gelatine). As you did not set the icing in any way, the fat from the butter slowly worked itself into the formed hooks of the proteins, destroying the mesh and returning the eggwhites to their original form (a suspension of proteins in water). Eventually, if you overbeat the meringue a bit, you also got some weeping caused by the proteins slowly getting tight enough to expell some water from the structure, but if present at all, this is probably a secondary effect in this case.
Bottom line: the mix you made is inherently unstable. I wouldn't use it as an icing unless the food is eaten quickly. You should get a different recipe. If pure buttercream is too rich for your taste, you can try versions which add yolks or whole eggs (not foamed eggwhites). The lecithine in the yolks emulsifies the rest of the egg with the butter, so you don't get separation problems. Another option is to forego buttercream completely and use whipped cream, it still gives you a rich mouthfeel, but has much less fat per weight and whips to a higher volume.
Best Answer
I suggest you use royal icing instead of butter icing. Basically made from icing sugar (powder sugar) and egg white, it will cost very little and can still be coloured if necessary.
Once you have tried out all the simple styles you scrape it off the board or work surface, mix it well and practice again. Store in a sealed air-tight container in the fridge for 1-2 days. This will give you time to make more intricate flowers, trellis, run out collars, nests, baskets, etc. on small squares of greaseproof paper, nails, etc.
Dry in an airy place and once completely set and hard, you can keep the results in a sealed container in a cool dry place for weeks to use as cake or cookie decorations. If you are really serious about becoming a sugar artist, it will be important to master royal icing, so you might as well start out with using it for "basic training", so to speak.