STR is used for Basic Melee Attack. Always.
DEX is used for Basic Ranged Attack with light thrown and projectile weapons.
STR is used for Basic Ranged Attack with heavy thrown weapons.
INT is used for Basic Ranged Attack for Wizard.
CHA is used for Basic Ranged Attack for Warlock.
Most (all?) Rogue attack exploits are keyed off of DEX.
All Ranger melee attack exploits are keyed off of STR.
All Fighter attack exploits are keyed off of STR, but a few (no at-wills or 1st level) allow you to add your DEX bonus to both attack and damage (damage only currently on compendium).
A high DEX low STR character is best realized as a Rogue, but only on your turn. (This answers the question of whether you can build a "sub-optimal" character in 4e: high DEX, low STR fighter specializing in Rapier vs. high STR fighter specializing in Longsword).