I wash clothes most evenings when I travel (not just biking), and it really cuts down on what you have to haul. Here are my techniques for getting almost everything dry by morning: 1. Don't put off doing the laundry until bedtime. Do it while the sun's still up if possible. 2. Wring it out the regular way, then roll it in a towel and wring it again. The towel absorbs a lot of dampness. 3. Hang it directly in front of the air conditioner or heater, and put the fan on. This usually involves moving a chair. 4. Check it before you go to bed, and put the wettest parts where they're more likely to dry. 5. Use the hairdryer in the morning, if something's still damp.
There are two things that might not dry by morning if it's damp, and one of them is bike shorts. So bring an extra pair, just in case. (The other is blue jeans.)
I love Fels Naptha soap for travel. It comes in a bar, so there's no mess, and it does a great job. I use it at home for stains.