Okay, so we got Initiative Passes and Combat Turns. In an Initiative Pass, you are only allowed 1 offensive action. Melee also is Complex to begin with, so doesn't matter. You can take the Multiple Attacks action, but that involves splitting the dicepool. So there's 2 things you need to execute a lot of attacks.
1: High dicepool count, so the split doesn't hurt too much.
2: High initiative.
Now sure there's initiative-boosting powers and gear, but a cheaper way is Edge. Just grab whatever Intuition and Reaction boosts you can get (Mystic Adept with combat-stats and Quickening for Increase Attribute spells is the most solid way but requires at least 8 karma after chargen and 7 kept during, an alternative is Resources A, Attributes B, +2 very expensive Intuition, +3 Reaction ware), then spend Edge to roll 5 Initiative dice.