This question is about as fraught as questions about Vietnamese immigrants and nail salons, or Indian immigrants and convenience stores. There are (partial) answers, but it involves walking through a mine field of after the fact-ism and folk history.

It seems that the stereotype had its seed planted in gold rush era San Francisco. There was a point in the 1850s that dirty laundry was being shipped by boat to other ports to be cleaned, many times at least as far away as Hawaii.^1 During the same period, there was a wave of immigration from China, either to escape the Taiping Rebellion or for work.

The Chinese Laundryman: A Study of Social Isolation by Paul Siu and John Kuo Wei Tchen credits Wah Lee for starting the first Chinese laundry in San Fransisco in 1851. Rather than laundry being expensively shipped, cleaned, and shipped back over the course of 6-8 weeks, residents could simply take it to a local laundry that did the job, for less money, faster. Over the next 25 years, the number of Chinese laundries in San Francisco alone shot up to 500, not counting all of the rest in other parts of California.

None of that directly answers why that expansion grew into a modern stereotype, but hopefully it partially answers how and when Chinese immigrants got into the clothes-washing business.

1: There's a really fascinating tangent here about gender roles in gold rush era San Francisco and the economics of shipping dirty shirts to The Sandwich Isles, but that's not the question at hand.