January, if fluffy chicks are your priority. Eggs hatch around Christmas, gentoo incubation runs about 34 days, and by mid January the colonies are full of the fat grey meatballs people picture. Aim at the first three weeks of January for the best odds. Hatching drifts by a couple of weeks year to year, so nobody can promise you a specific stage on a specific date.
The rest of the progression is worth knowing, because later months aren't the write-off people assume.
February. Chicks are off the nest, muddy, and being chased around by parents. First swims happen late in the month.
March. Fledging, and leopard seals hunting chicks as they take their first swims. Dramatic, if that's what you're after.
April. Fewer penguins, but they're still there. A first timer wouldn't notice the difference.
If baby penguins are the whole reason you're going and January doesn't work, South Georgia is the answer. King penguins run a different breeding cycle, so there are chicks there right through the South Georgia visitor season.