It's actually quite easy to raise a caterpillar that you have collected in the fall. The key is to provide conditions that mimic the caterpillar's natural cycle and habitat. Some caterpillars survive the winter by simply burrowing under the leaf litter, or squeezing into bark crevices. Others pupate as cooler weather approaches, and remain in this state until spring.
If your caterpillar is still feeding, you will need to provide food as you would for caterpillars you find at other times of the year. Eventually, the caterpillar will stop feeding and may become more sluggish; this is a sign that it is preparing itself for winter.
For species that overwinter as caterpillars, simply clean any remaining frass and food plants from the container, and then cover the resting caterpillar with a layer of leaves. Move the container to a porch or unheated garage or shed for the winter months. When spring arrives, watch for signs of activity from the caterpillar.
Butterfly caterpillars often overwinter as chrysalides. Provide some twigs or stems so the caterpillar can suspend itself to pupate. You can secure the twigs with clay at the bottom, or cut pieces that will fit tightly against the container without falling. Once the caterpillar has pupated, move the container to an unheated area for the winter.
Moth caterpillars typically pupate in the soil, sometimes incorporating leaves into their pupal cases. Place a layer of peat moss in the container, and add some leaves. Once the caterpillar spins a cocoon, you can remove any remaining leaves and move the container to an outdoor or unheated location.
If you are storing your pupae or caterpillars outdoors, be sure to keep them out of the sun. Even on a cold winter day, the container can warm up quite a bit if placed directly in the sun's rays. This might lead to premature emergence, or may cause the pupae to dry out.
When spring returns, you should keep your caterpillar or pupa cool until other members of the same species are emerging. If you aren't sure, wait until the trees in your area start to leaf out before moving the container to a warmer location.