Though Victoria Falls is technically located in the South African country of Zimbabwe, Victoria Falls National Park (or Mosi-oa-Tunya National Park) is to be found on the southernmost border of Zambia. It is the falls and the Zambezi River which separate the two countries. To enter Zambia, both a passport and a visa are required. You may purchase a visa on arrival for $25. (British passport holders -cost £35.  You must pay in Sterling.  It is important to carry the exact amount with you, as change may not be available.  For further information on Zambian visa requirements you should contact:  Zambian representation in the UK)

Be sure to have the original passport or a certified copy on your person at all times and, when departing, have an extra $20 handy to pay a departure tax. Air travel is usually by way of Lusaka International Airport, which serves several airlines including British Airways, Air France, and South African Airways. You can then book a flight on one of several in-country air charter companies to carry you on to your destination.  

But that's not your only option. Though perhaps less convenient, a bus can get you there If you are renting a vehicle, plan to have a temporary import permit or a "use of vehicle permit." Road checks are common, so keep any such papers handy.

By far, the most adventurous way to get to Victoria Falls is probably by train on Zambia Railways or Tanzania-Zambia Railway. Get a ride to Victoria Falls (the town) in Zimbabwe, and then take an excursion train across the bridge (don't look down) to Livingstone and back again.