Peak varies with elevation, rainfall during the summer and fall weather conditions. It can range by a couple of weeks from year to year. The higher elevations are (1) predominately coniferous and so less colorful (no less beautiful - ground vegetation colors in golds and magenta)(2)have leaf fall relatively early (late Sept).
Mid-October is often past peak in the Whites. The first week of October should be great - maybe a little past peak but still lovely. Check here to get a sense of timing:
Others have given good advice on where to go and I third or forth the observation that staying low will maximize the color.
The Pemi is a great option that allows loops, alpine travel under the right conditions, lots of hardwoods, a real mix of easy and strenuous hiking - options galore.
The Kilkenny really isn't large enough to take 4 nights in unless you REALLY like to dawdle. If that is the case, you could stay in the Cabot cabin one night, Unknown Pond, and Rogers Ledge... that's a 2 or 3 mile day... and then out.
It's also possible to link the pemi and the sandwich range - but it requires some road walking.
The carter mariah range taking advantage of the shuttle service that the AMC provides from Pinkham Notch
Lovely and not crowded - loops are possible across the wild river basin and along the royce > baldface ridge
Great overview map of the entire area:
this is heavier than the AMC alternatives but covers ALL of your options for planning purposes and provides PLENTY of detail for hiking