That depends on the caterpillar, but given the
limits caused by the weather, I expect that most
caterpillars will grow fastest either in the
spring (California) or summer (most of the rest of
the U.S.).
Answer 2:
In general caterpillars grow the fastest when
the weather is the warmest. This tends to be in
the summer, which is also when most butterfly/moth
breeding cycles occur.