This is where modern fabrics come into their own - in normal life I stick to the natural materials rule, but for the Camino I raided charity shops for all the lightweight polyester hiking gear I could find. I found it well worth road-testing everything - to make sure that trousers and shirts didn't cause chafing when sweaty, and would dry overnight when handwashed. I took a concentrated laundry soap, but it isn't really necessary - and as suggested, a combined shampoo and shower gel can be used for everything.
You can wash yourself and your clothes in the shower, wring them out well, then after drying yourself use your towel to roll them up and wring them out again.
Socks - both wool and liners (and occasionally my towel) usually needed to be pinned to my pack the next day to fully dry - on the advice of forum members, I took a length of elastic that went round my pack and acted as a washing line, and used diaper/nappy pins to secure everything.
You can easily get by with one change of clothes, but I was happier taking 3 lots of socks and underwear - and did take an extra smartish pair of hiking trousers and a shirt for going out in the evening, and as backup in case of rain. Even with wet weather gear, cold weather gear (for early May) lightweight sandals, a pair of 'going out' soft leather shoes and everything else - it was still possible to weigh in under 6.5Kg. And it all came from charity shops - but took a fair amount of obsessive packing, repacking, weighing and discarding. But I will probably take all the same stuff again next time, as it all worked well.
All part of the fun!
There was a ad for Rohan gear (in the early 80's?) that I loved, with an endorsement from some mountaineer along the lines of: 'you can descend from the summit, wash your clothes in a mountain stream, dry them on a rock in a jiffy, and it's drinks at the Embassy at six!'
That's the sort of gear you need...