What you are requesting, by my understanding, is for the fountain to power itself. This is not feasible.
Think of the water as a ball. The ball falls down, and as it falls it is slowed down by air resistance. This is energy lost. The ball falls 10ft, and when it lands it has less energy than it did when it was 10ft. high, because it has lost some to air resistance.
This ball lands on a lever, pushing another ball of the same weight upwards. The second ball will also be slowed by air resistance, and will only reach 9ft. high.
When that ball comes down, it launches the other ball 8ft. Then 7, 6, 5, and eventually none. Because energy is lost.
To will need to put energy in from another source to boost the energy the falling ball is exerting so that the second ball can make it back up by 10ft.With regards to your fountain, this means that the falling water will never give you enough energy to raise the same amount of water back up, because some of that energy is lost to the surrounding air. You can probably get something of an idea for this by using a water bottle and a ladle. Pour the water into the ladle, running down the inside edge so that it is redirected upwards. The water coming back up will never reach the bottle you're pouring it from.
So, you will need a power source. You will need to decide what suits your application best (solar, battery, mains, fire, water wheel, wind turbine, exercise bike, geothermal pylon, nuclear reactor, etc.) and then design your fountain with this in mind to power some sort of pump to move the water upwards.
Alternatively, you could build some form of reservoir to harvest rainwater and keep your fountain stocked with water that it higher than it. At that point, you're making something close to a natural waterfall! Provided your reservoir is large enough to carry you through the lowest average rainfall for your area, the fountain will continue to run all year round.