So, my idea for my upcoming rogue character is really inspired by Edward Kenway. I planned on using two rapiers as my main weapons, and dual wielding them. However, I am not sure if I can do that if I'm bound by normal rules, or how it would work if it even could work. I am not a particularly experienced D&D player, so if there's anyone with knowledge over this, please help me understand it all.

How 'dual wielding' works is that you can hold any two weapons you like that don't have the 'Two-handed' property.  A wizard can hold both a longsword and a staff without an issue, and your rogue could hold two rapiers without issue.  To attack with two weapons in one turn, you use the Two Weapon Fighting rules, which states that if both of the one-handed weapons you're using have the 'Light' weapon property, you can use the Attack action to attack with the one in your main hand however many times your class features lets you attack.  Then, as a bonus action, you can make an attack with the weapon in your off hand, but you don't get to add your ability modifier to the damage of that attack.  I.e. if you had two handaxes in your hand as a level 5 Fighter with 16 Strength (+3), you could attack twice with the main handaxe and deal 1d6+3 damage with it.  Then, provided you used the Attack action with your main thing, you could use a bonus action to attack with the off-hand handaxe for 1d6 damage, NO +3.

To do this with rapiers you would need the Dual Wielder feat, which eliminates the requirement for the weapons you're holding to be 'Light'.  To gain your ability modifier on the damage of the off-hand attack, you need to acquire the Dual Wielding fighting style from either the Fighter (level 1) or the Ranger (level 2).  Without the feat you can only attack with one rapier, no matter how many you're holding.  Without the fighting style all you ever do is the weapon's damage dice plus magical bonuses, no ability modifier.

Take a quick glance at the rules.
First there's the general ability to wield and attack with two light weapons: Two-Weapon Fighting. (Every class and race gets this automatically.)
Then there's the Dual Wielder feat that removes the need to use Light weapons.
Finally there's the Two-Weapon Fighting Fighting Style from some classes (Fighter and Ranger like Yurei said) which allows you to add your ability modifier to the Bonus Action attack.

In order to be able to attack with two Rapiers you will need the Dual Wielder feat because they are not light. The Fighting Style is certainly optional - but you'll probably want it.

Using two shortswords will allow you to get two attacks without a feat or fighting style. Your primary (main hand) attack will have the full damage bonus and the secondary bonus attack (dual wield) will not include the damage bonus. Either one can trigger sneak attack as a rogue, though. (If the main attack misses and the bonus attack hits, for example.)

Kenway used something similar to a modified cutlass from what I can tell. In 5e, swords are mostly limited to longsword, rapier, shortsword, or scimitar. But given that shortswords are finesse, light, and piercing, they fit the bill as capable of dual wielding swords. If you Google "smallsword" you'll get some real life examples. Seem somewhat similar to what Kenway might have used. 

From what I've read so far is that I will be able to dual wield rapiers once I get Dual Wielder as a feat, and then I should multiclass with Fighter in order to get the Two-Weapon Fighting Fighting style to get ability mod to both attacks, is that right? Without the Fighting Style, am I still able to use the modifier to my hit roll?

You'd still get the to-hit modifier, yes.  Just no damage.

I would talk to your DM about reskinning shortswords as smallswords.  As Sigvard said, google 'smallsword' and you'll see they look mostly like scaled-down rapiers.  Historical rapiers are actually very big things and do not dual-wield well at all.  A smallsword would be an easy enough reskin for shortsword though and provide the same feel without having to muck with the otherwise kind of butt Dual Wielder feat.  Especially on a Rogue where you're not getting a ton of benefit from it.

If D&D wants to pretend there's only five kinds of sword in the world, players have every right to take some of the hundreds of other swords that actually exist and shove them back in.

I thought of the rapier concept too but then the more I looked into it, I saw that dual wield is a rater awful feat. By dnd 5e rules, you only apply your strength/dexterity mod to one of the two attacks meaning that you would do similar if not the same damage as you would with a two-handed weapon. 

This doesn't seem to be a problem with the rapiers since the 2d8+dex would still be better than the strongest (normal) two-hand weapon to choose from, the greatsword, that deals 2d6+str. However, once you get to fifth level you get an extra attack which gives two-handed weapons the MAJOR advantage. with the extra attack, you only get one extra onhand attack with two-weapon fighting, while the two-handed weapon wielders would get a second entire attack. The two-weapon fighting becomes just as good damage-wise (3d8+(dex)2 vs. 4v6+(str)2). While yes they still have the same damage max you have to take a full-round attack to meet the same damage as a two-hander can do in one action.

There is a way however that the DM can make the fighting style just a little better. If the DM allows you to attack with both hands in the same action then this easily evens out the playing field, so before you chose two-weapon fighting, you might want to consult with your DM about making this exception.

The OP, two years ago, was talking about a rogue. Rogues can’t use a greatsword, if they could, they couldn’t sneak attack with it, and don’t get a second attack at level 5. For a rogue, really, dual wielding is about getting a second chance to trigger a sneak attack if you miss with your first attack. Of course, that ends up conflicting with your bonus action to disengage and get out of there.

I agree that dual wielding is generally sub-par, but for a rogue, the reasons are different than the ones you list.