Hard:
Complete Normal

Dante Must Die:
Complete Hard mode
Heaven or Hell:
Complete Dante Must Die

Quicksilver style:
Defeat Greyon at the end of mission 12

Doppleganger style:
Defeat Shadow enemy at the end of mission 17
Gallery:
Complete Easy or Normal mode
Gallery update:
Complete Hard

Unlock Costumes and Characters:
Press R1 or L1 to select a different outfit or character at the Mission select screen.

Coatless Dante:
Complete either Easy or Normal mode
DMC1 Dante:
Complete Normal mode
Coatless DMC1 Dante:
Complete Hard mode

Super Dante:
Complete Dante Must Die
Sparda:
Complete Hard
Legendary Dark Knight:
Clear Hard Mode
Cerberus:
Defeat Cerberus at the end of mission 3
Agni & Rudra:
Defeat Agni and Rudra at the end of mission 5

Nevan:
Defeat Nevan at the end of mission 9

Beowulf:
Defeat Beowulf at the end of mission 11, and then pick the weapon up at the start of mission 14

Force Edge:
Force Edge is usable with Coatless DMC1 Dante, DMC1 Dante and Legendary Dark Knight after unlocking them

Yamato Yamato:
Yamato is only usable when controlling Vergil in the boss fight in mission 19 after Dante is grabbed by the boss.
Devil Arms:
Devil Arms are obtained by defeating certain bosses at various points in missions and are usable with a certain character or costume.
Extra Ending:
To get a special screen kill over 100 enemies during the ending credits.
Firearms
Get fireamrs in certain places or by defeating particular bosses during the game.
Shotgun:
The shotgun is on the wall of the front bar in mission 3.

Artemis:
Found after inserting the last item into the statue in the Mute Goddess statue in mission 6.

Spiral:
The Spiral is found on the platform in the Subterranean Lake in mission 9.

Kalina Ann:
You get from Lady after defeating her in mission 16.
Unlock ALL Costumes, Modes and Gallery (Japanese Version):
Hold down L1, L2, R1, R2 at the Start menu and rotate the left analog stick until you hear Devil May Cry.