The trick with most pump stacks is how to power them. As you've observed it's easy to set up a series of screw pumps to move liquid up a number of z-levels - there's a couple of interesting bits to them though.
The wiki's design has the floorplan laid out as follows for the 'intermediate' pumps;
OOOOOO
O+_+_O
OOOOOO
Legend: O=wall, +=floor, _=channel
In alternating layers of course, design mirrored for successive levels. This leaves each pump in a 'Hanging' state in contact with the one beneath it - this allows it to transmit power. Of course this also makes it difficult to build and maintain - any pump built with no support will deconstruct (possibly with other consequences), so you need to build up from a solid base or provide support through adjacent gear assemblies or axles. Also, building destroyers can sabotage this design by dismantling any single pump.
If you want to go the extra mile and make it self-powering... your reservoir will need to drain such that it always has moving water of appropriate depth under the water wheels. Oh, and obviously you'll need waterwheels too. The catch is that even if you manage to construct an internal powering system it'll need to be started somehow, which will involve a separate power source (in which case you wouldn't need the self-powered bit so much), or dwarves on the pumps (which has its own problems, naturally, especially if there's no access to the system or if it is an accidental drowning trap).
It'd probably be
safer easier to construct either a small Dwarven Water Reactor or, if you are blessed/cursed with an aquifer, to use that as a power supply through the infinite aquifer power trick detailed on the forum.
Dwarven Water Reactor
z=1
OOOOO
OW_WO
OW%WO
OW%WO
O+_+O
OOOOO
z=0
OOOOO
O777O
O7O7O
O7O7O
OO7OO
OOOOO
Infinite Aquifer Power trick
z=1 (Above aquifer)
OOOOO
O__WO
O__W*
O__WO
OO%OO
OO%OO
OO_OO
OOOOO
z=0 is channeled-out aquifer beneath channeled tiles
Activate by using pump once - make as big as necessary
- pump access not shown but can be from anywhere
Legend: O = wall
% = pump
_ = channel
+ = floor
7 = water
* = mechanism / axle for power output
Whichever method you want to follow, good luck and remember that if it doesn't work you could modify it into a drowning trap with minimal effort - if it wasn't already, that is