All characters can effectively dual wield light weapons.
All characters can effectively dual wield and fight with two weapons so long as both of the weapons being held have the light property. The Player's Handbook describes how this works on page 195.
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. (PHB, 195)
There is no need for any feat, skill, or class feature to do so. This is a base action that all characters can use.
I assume your DM is referring to the feat Dual Wielder. This only is necessary if you are attempting to dual wield and attack with a non-light property weapon. For example, if you wished to dual wield two longswords, you would need to have the Dual Wielder feat.
As for if your DM is correct, well, that depends on what weapons you are attempting to dual wield. If you are attempting to dual wield and attack with two light weapons, he is wrong and you may do so with no feats or skills. If you are attempting to dual wield and one of your weapons does not have the light property, he is correct and you must acquire the Dual Wielder feat first.
Warlocks do get bonus actions.
All characters get bonus actions. There is no rule in the official books that states warlocks do not get bonus actions. This is described on page 189 of the Player's Handbook.
Various class features, spells, and other abilities let you take an additional action on your turn called a bonus action. ... You can take a bonus action only when a special ability, spell, or other feature of the game states that you can do something as a bonus action. (PHB, 189)
Since the Two-Weapon Fighting feature states that "you can use a bonus action", you get a bonus action. Your class does not matter.
If your DM continues to insist that Warlocks do not get bonus actions, ask them to explain how you use the Hexblade ability Hexblade's Curse, which requires you to use a bonus action.
In regards to your edited addition:
Hexblade dual wielding is not different.
Dual wielding is the same for you as everyone else. It does not change what modifiers you use. If you have the ability to use Charisma as your attack stat with melee weapons, you can continue to do so while dual wielding.