Happy Easter! and good Passover!  For a short video about our Savior's role, and what this week means to us, see His Sacred Name: An Easter Declaration.  I'm grateful to have this holiday as a reminder of His life and willingness to do our Heavenly Father's will.  Eggs remind us of the promise of life from something that looks only like lifeless, tomb-like stone.
The yellow eggs were dyed using curry powder, the orange ones used onion skins.

When you dye eggs using natural colors, put water, a little vinegar, the eggs, and your 'dye' in a pan.  Boil to both cook the eggs and set the color. (See here to get The Perfect Hardboiled Egg.)  The colors tend to be softer and often muted- you won't get stop-sign-red anywhere.  Each egg ends up a little bit unique, though.  That's part of the beauty.  The vinegar helps the dye penetrate and set.  Use about 1 Tbsp for every 2-3 cups of water.

I'd heard that eggs should be washed first, to prepare the shells to accept the dyes.  I tried it and noticed no appreciable difference between the washed and unwashed eggs.  So save yourself the time.  Use eggs straight from the carton.  Or chicken, if you're so fortunate.  (NOTE:  the older the egg is, the easier it wil peel!)

For cool effects:
Before cooking/dyeing
-draw on it with crayons
-wrap it with rubber bands or yarn
-dab, wipe, or drizzle on a little oil or melted wax.
-put 1 Tbsp. oil in the cooking/dyeing water.  This will make some marbled irregularities on the eggs.

After cooking/dyeing:
-sprinkle with salt while still wet
-for a more marbled or mottled egg, rub it a little while it's still warm.  Or pat/rub with paper towels. 
-let the eggs cool while still draped, wrapped, or sprinkled with what you used to dye it.  (See photos.)
-partially- or completely- submerge in another dye.  
-draw on it with markers when dry

To make it gleam, rub a little vegetable oil on the egg after it has cooled.

Ideas for dyes:
Blue to Purple- grape juice, red grapes, blueberries, red cabbage, blackberries, violets
Brown- white oak bark, juniper berry, coffe, barberry twigs, black tea
Buff- crushed walnut shells, green tea
Golden Orange- onionskins (yellow onions)
Orange- turmeric, ground cumin, onionskins plus beet juice
Mauve- grape juice
Green- spinach, kale, alfalfa, parsley, carrot tops.  Or dye an egg yellow then dip in blue dye.
Pink- beet juice- let eggs sit for 30 minutes in it (liquid from a can of beets, or from cooking them)
Reddish Purple- paprika, skins from red onions; beets- boil in the juice, shredded, sliced, or chopped Yellow- Boil with 1-3 Tbsp. turmeric or curry powder, or dye when already cooked- use 1 tsp. turmeric and 1/4 tsp. vinegar in 2/3 c. boiling water
These were dyed using 1 Tbsp. curry powder in just enough water to cover the eggs. The two lighter eggs were pulled out immediately when fully cooked. The others were cooled in the colored water, with a tray of ice cubes added to help them stop cooking.

Fill the pan with eggs, stuff it as full as you can with your dye (onionskins, here), and bring it to a boil.
If you're using a spice or a powder, use 1-3 Tbsp.

For comparison:
Clockwise, starting with upper left: (1) boiled and drained to cool; (2) boiled and covered with wet onionskins while cooling, (3) boiled and left in the hot dye until completely cool, (5) plain white store egg, (6) plain brown store egg.

Each egg ends up a little marbled, a little unique.  I  find them fascinating to look at.

Any place your fingers, spoon, pan, paper towels, or other eggs touch, the dye will be lighter.