Does the capitalist buy the raw materials that make up a commodity for less than he sells the commodity?
Yes, except materials aren't conscious and alive so it doesn't matter how much you buy and sell them for.
How do the raw materials get from being worth less (just raw materials) to worth more (the end commodity)
Demand from people. People want the commodity more than they want the raw material. People want milk in a container over their own personal cow.
If the labour caused the raw mats to go from worth less to worth more
Not just that labour, but the machinery. In fact, machinery is ever increasingly taking a larger portion of the pie with the rise of automation. The machinery comes from the capitalists or investors. The labour cannot turn a pile of raw material into a car by themselves, they require the use of machines. This also usually requires an organisational value to bring the labour, raw material and machinery together.
Value added by worker = End price - Raw materials
*Labour Value + Machinery Value + Organisational Value = End Price - Raw Materials - Rent - Bills - Wages
labourer earned what they added to the commodity the capitalist would not profit and would break even
See the more accurate above formula.
What the capitalist/investor takes is subjective but is less subjective or more justified than your post implies.