In order to acquire the Leviathan Summon Materia, you'll first have to clear Chadley's Battle Intel Reports 01 through 14 — including Shiva and Fat Chocobo. This will unlock the VR battle.
Leviathan — Boss Fight Strategy
This fight comes in two distinct phases. Before you set out, prepare yourself well.
Because Leviathan has a lot of AoE abilties, make sure you have Healing Materia + Magnify Materia set up on your primary healer (usually Aerith). The more levels in Magnify the better, for stronger heals. Pray Materia is also a good option for a secondary healer, in case of a bind, as is stocking up on Mega-Potions.
Phase 1 Strategy
In the first phase, Leviathan can use any of its attacks, including the ultimate Tidal Wave — which you particularly need to watch out for.
The main thing here is to try and stick close to the serpent, and also to group up as tightly as you can so heals are easier. This will also cause his Briny Bellow attack to miss, opening it up to being attacked hard.
At some point, usually around half health, Leviathan will use Tidal Roar and fly up in to the air, which begins Phase 2.
Phase 2 Strategy
Tidal Roar also releases water spouts on the battlefield. Switch to Barret here, and use spells to do as much damage as possible — you may even get a Stagger here.
Be mindful that these spouts also block attacks if they happen to be in front of Leviathan, so avoid attacks like Focused Thrust or Overcharge if so.
For most of Leviathan's attacks, you can block when the name appears above its head. The animations take quite a long time to get through, so you have plenty of warning.
When you see Tidal Wave appear above Leviathan, make sure your characters heal as it can hit hard! If you can Stagger it beforehand, great — if not, eat the attack, heal up, and keep plugging away.
Eventually, Leviathan will go down, earning you another piece of Summon Materia!