When we milk, we use a bucket that has two compartments. One compartment has hot water, a bit of dish soap plus 4 drops of tea tree oil.
The second compartment has just water and 4 drops of tea tree oil. Each compartment gets its own little towel.
I thoroughly wash the udder and teats with the first water and towel from the first compartment, then using the towel and water from the second compartment I clean everything off.
Doing this delivers a squeaky clean udder, and we've never dealt with mastitis mid-lactation.*
* We've dealt with mastitis, but it came on directly after calving due to various complications and wasn't a result of unsanitary conditions.