It's possible to buy special kits to dye eggs for easter but here is
a very simple method that produces beautiful eggs and works with
standard liquid food coloring from the supermarket.
For best results I recommend to use white eggs.
Hard boil the eggs.
Prepare a solution to dip the eggs into: Fill into a large glass bowl or old pickle jar 1/4 liter of water, 1 table spoon
of vinegar (5%), 8 drops of food coloring. Note that food coloring
stains cloth, tables and kitchen counters. It's a good idea to put
a news paper on the table to protect it.
Set the eggs into the prepared dye solution and let them dye for 1-4 hours dependent on how strong you want the colors to be.
Prepare a large bowl with crumpled up news paper. This is to dry the eggs.
Take the eggs out of the jars with a large table spoon and set them on to the crumpled up news paper to let them dry.
Boiled eggs must be stored in the fridge and they keep for a week and probably longer (I have not tested how long they keep).