This article is from the Final-Fantasy and other Square Soft Games FAQ, by firstname.lastname@example.org (Nick Zitzmann) with numerous contributions by others.
First you must have gotten Cloud's Ultima Weapon, and he must know all
his Limit Breaks except for his Extreme Limit Break. Go to the Gold
Saucer during Disc 2 or 3 and collect enough Battle Points so that you
can buy either the Omnislash Manual or the W-Summon materia. Purchase
one or the other, then collect more points and buy the remaining item.
Use the manual so that Cloud will learn his Level 4 Limit Break, then,
go and talk to the receptionist, just as you would if you wanted to
fight a normal battle. Instead, you'll be given the option to partake in
Dio's secret battle against his hand-picked enemies (all bosses, I
believe). Only Cloud can fight the 'secret' battle. Depending on how
well he does, he can get the Combat Diary, Gambler, or Autograph items,
and if he wins all eight battles, he'll be given the 'Final Attack'