Purchase or find in your garden a swallowtail egg, attached to a leaf, or a small swallowtail caterpillar. Different species prefer different plants for their food source. A spicebush swallowtail, for instance, will only lay its eggs on the leaves of the spicebush plant. Tiger swallowtails like willows, ash, cherry, birch, tulip and cottonwood trees as well as lilac bushes. A tiger swallowtail caterpillar is a surprise waiting to happen because you can't predict what color it will be when it emerges.
Attach the tiger swallowtail egg, on its leaf, to a twig in a see-through container. Make sure there is sufficient air circulation and sunlight so the container doesn't develop mold. Collect or purchase a large quantity of fresh, pesticide-free leaves from one of the plants tiger swallowtails eat in their caterpillar stage. Keep the leaves fresh in a plastic vegetable bag in the refrigerator crisper. Have a ready source for more fresh leaves available, in case you run out.
Watch the caterpillar crawl out of its egg. It will immediately begin eating, probably the discarded shell. Then it will turn to leaves. Your job now becomes keeping the caterpillar well fed with fresh leaves. At this stage, the tiger swallowtail has the protective coloration of bird droppings. It will be a small brown squiggle with a white band or white markings on it --- not visually appealing to hungry birds.
Check leaves for freshness daily. Leaves that get old and tough cannot be eaten by the caterpillar, and it will starve without enough edible food. As it grows, your caterpillar will turn plump and green. To do this, it needs to shed its old skin several times. Don't be alarmed if you see this happening and if the caterpillar eats the old skin.
Provide some leaves still attached to twigs or clip or tie a few leaves to a twig so the fat green caterpillar can "tent" itself. It will make a hidden nest by pulling a leaf around itself so it is invisible. It fastens the leaf with silk threads it spins. Once the caterpillar is large, smooth and green, it will show a different camouflage: yellow eyespots with black "pupils" on its head. It will also have a yellow band around its "neck" and look a bit like a small snake to a bird -- another good reason not to eat it.
Keep fresh leaves in the caterpillar container in a moist but not wet or damp environment in a sunny but not hot or drying spot. The caterpillar will eat maniacally until it is approximately 2 inches long and then stop. It will probably turn brown.
Don't touch the fragile chrysalis as your caterpillar envelopes itself in a brown shell and hangs by thread it spins from a twig or sometimes the side of the container. Monitor the chrysalis until you see striped wings through it and get ready for the butterfly to emerge. The insect will be creamy white or light yellow with stripes, or striped and dark. Dark tiger swallowtails resemble a poisonous insect that sickens birds.
Place a shallow dish with some sugar-sweetened fruit punch or other sugary drink in the container. The butterfly will sit for a long time as its wings harden before it tries to fly. Then it will look for nectar, the sugar drink. Once the wings have hardened and it is eating, you may release it in the garden, near a favorite tree in leaf or a lilac or butterfly bush.