How do you create a waterfall elevator?

Solution 1:

If I understand the theory, here's how you do it.

≋≋≋≋≋█ ← remove this           - ground level
██████   when ready
     █   
     █   
     █  
 ................................................... 
     █   
     █   
     █   
     █       █                 - mine level
      ███████

Remove the marked block and this happens:

≋≋≋≋≋≋~                        - ground level
██████≋              
     █≋  
     █≋  
     █≋ 
 ................................................... 
     █≋  
     █≋  
     █≋  
     █≋~~~~~~█                 - mine level
      ███████

To move up and down, swim up and down the column of water; move slightly out to breathe then continue your ascension.

A boat makes this faster as Strix explains and this video demostrates. However, you must allow further space down to avoid crashing on the bottom:

 ................................................... 
     █≋  
     █≋  
     █≋  
     █≋≋≋≋≋≋≋█                 - mine level
     █≋≋≋≋≋≋█
     █≋≋≋≋≋█                   - extra space to break the fall
      █████

This video shows, however, that you can also climb a boat elevator by simply swimming.

Solution 2:

Boats float in water. Essentially, all you need is a waterfall from ground level straight down from the top of your mine shaft to the bottom. Then, while at the bottom, boat your way into the waterfall and watch as you rocket to the top.