You can say you're a duelist without being a duelist

It's entirely possible that it's the idea of a duelist that's appealing not the what the prestige class duelist actually does. You might be imagining Captain Blood, Robin Hood, or the Dread Pirate Roberts (or Prince Humperdinck who's probably actually a ranger).¹ Maybe the character you're imagining when fighting leaps around the battlefield, quipping and stabbing, and when not fighting scouts, tracks, and investigates like a boss.

That's laudable, and that can be pursued as a ranger without any need for the duelist prestige class. The character's ability scores even support that. Most characters don't take the feats Power Attack (with its Str 13 prerequisite) and Weapon Finesse. This character can, and the feat Power Attack will make your damage competitive (although you probably won't hit very often). Add to that things like the typical ranger benefits of bigger favored enemy and terrain bonuses, appropriate magic weapons, wands of ranger spells, and the feat Boon Companion to get your bird up to par, and you can have a perfectly serviceable guy-who-calls-himself-a-duelist without looking at the duelist class at all. Most rangers aren't toting a Charisma of 16, and duelist does nothing with that, but a ranger can. You can play this character for fun, worrying about things like optimizing your damage per round or whatever when you've a better feel for the game.

But it's possible there's something about the duelist prestige class that you must have. If that's a thing:

Going from ranger to duelist is unpleasant...

The prestige class duelist has as requirements a base attack bonus of +6, 2 ranks in each of the skills Acrobatics and Perform, and the feats Dodge, Mobility, and Weapon Finesse. Only the ranger's base attack bonus encourages him to enter duelist; literally everything else discourages a ranger taking levels in duelist: the skills aren't ranger class skills and the feats aren't available as ranger bonus feats.^2,3

This is especially true of the character described above. The feat Weapon Finesse, while a requirement for duelist, gets the character but a +1 bonus on attack rolls, enabling the use of his Dex 15 instead of his Str 13 but thereafter vastly limiting the character's weapon choices. The combat style two-weapon fighting doesn't mesh with the duelist class feature precise strike, which mandates the duelist not attack with a weapon in his off-hand.⁴ Further, The character's precise strike damage is limited by his Intelligence score. A ranger even gets medium armor proficiency and shield proficiency while duelist mandates he use neither medium armor nor a shield to get the benefit of the class feature canny defense.

This doesn't mean this character can't go duelist, but it does mean a ranger like this just isn't that good at being what the game imagines the duelist represents.

Also, be aware that you'll never see the best abilities the duelist gets if the campaign ends at level 12.⁵

...But you can make it work if you must

If duelist is important, there's nothing the character can do to enter before character level 7. There's just no way to increase base attack bonus faster. At level 3 take the feat Dodge and at level 5 Mobility. Stay ranger through those levels, at level 4 taking the companion hunter's bond rather than the animal companion (the companion bond is terrible but the animal companion is worse after multiclassing even with the feat Boon Companion), at level 5 getting a 2nd favored enemy, and at level 6 probably reluctantly taking the two-weapon fighting combat style bonus feat Two-weapon Fighting. At level 7 take the feat Power Attack; it'll make your damage-per-round competitive.⁶

After that, get a keen rapier as soon as you can and start crit fishing—i.e. bombarding your foe with a series of small attacks in hopes of critical hits that will often burden him with status conditions instead of killing him outright. Take at level 9 the feat Critical Focus, at level 11 Destroy Identity if you can, at level 13 the feat Amateur Swashbuckler and add to your weapon the weapon special ability skewering, and at level 15 Blinding Critical because Pathfinder makes getting alternate senses like blindsight or tremorsense difficult, and protecting oneself from getting one's eyes gouged out is really hard to do.

Alternatively, take at level 9 the feat Improved Critical, at level 11 Critical Focus, at level 13 Destroy Identity if you can, and at level 15 Blinding Critical. Confirm critical hits against evil creatures automatically using lots of oil of bless weapon (1st-level spell at caster level 2) (100 gp; 0 lbs.), skipping the keen, skewering rapier.⁷

Either way, you're essentially playing to the duelist's 10th-level class feature crippling critical, which this character may not even see and which goes unsupported by the rest of the prestige class unless counting the abilities combat reflexes (not the feat but the feat's benefits—sigh), parry, and riposte which, when combined, sometimes maybe lets you protect an ally while simultaneously stabbing a foe when it's not your turn... by skipping some of your attacks.

Finally, note that this is not flashy. Almost any character with a good base attack bonus could go this route. The warrior NPC class does this as well as this ranger. The reason the sample duelists start as fighters and rogues is that those classes have archetypes and class features that support becoming duelists. The fighter and rogue foundation is just firmer than the ranger one.

It's not too late to do something else

I really believe a player shouldn't be stuck with a disappointing character. I don't think games should tell players that it's okay for a character to suck now and be awesome later when another character gets to be awesome now and later. Given how frequently games collapse, dissolve, and implode, playing a character who's no fun to play because he can't contribute mechanically shouldn't even be an option, and all too frequently it is.⁸

Thus the best option would be to start over and pick things that will enable your character to do what you wanted your character to do in the first place. Honestly, the character's race, class, ability scores, and feats have very little synergy—you can try to hit hard, but you'll never hit as hard or as often as those folks whose jobs are to hit hard and often, even if (maybe especially if) your next feat is Power Attack. If hitting hard was your goal—if that's how you expected to have fun in this campaign with this character—that's probably just not going to happen.

Despite that, this character can still be entertaining and interesting in different ways. Below are two.

• Focus on the skill Diplomacy. At level 3 take the feat Additional Traits. Use one to get the Diplomacy skill as a class skill (I like the trait Extremely Fashionable.) Use the other trait for something awesome like Anatomist (the bonus isn't great but it works with advice above), Finding Haleen, Reactionary, or Second Chance. Also ask the DM if the character can change Skill Focus (Survival) to Skill Focus (Diplomacy), maybe by using the rules for retraining. This puts the character's high Charisma score to good use. The Diplomacy skill makes some GMs angry, though, because the character befriends creatures he's supposed to be killing.
• Focus on the special ability wild empathy. Pathfinder made the special ability wild empathy interesting... if a character's willing to invest in it. At level 3 the character can take the feat Fast Empathy, cutting the time to use wild empathy from 1 min. to 1 standard action. The feats Greater Wild Empathy and Vermin Heart (or a scarab of Khepri (Pyramid of the Sky Pharaoh 61) (7,800 gp; 0 lbs.)) expand the kinds of creatures wild empathy can affect. Then, when able, get a circlet of persuasion (4,500 gp; 0 lbs.). While not useful all the time, when wild empathy is useful, the character will likely win the day.

Seriously, though, my real recommendation is to have fun in this campaign. You can't really do the hobby wrong. Experiment and take risks. Worry about wacky and weird optimization stuff in the next campaign, keeping track of errors, omissions, house rules, likes, and dislikes during this one.

¹ According to Buttercup, Humperdinck can track a falcon on a cloudy day. I'm not entirely sure how anyone does that, but that's badass.
² The ranger combat style faithful (Inner Sea Combat 11) for the deities Besmara, Calistria, Cayden Cailean, Desna, Norgorber, and Zon-Kuthon permit as a bonus feat the feat Weapon Finesse; the deity Sarenrae permits as a bonus feat the feat Mobility. None permit both, and none permit the feat Dodge.
³ The class does nothing with the Perform skill. The character can put 2 ranks into Perform (keyboard instruments) or Perform (percussion instruments) to meet the requirement. When I think duelist, I think Elton John and Phil Collins.
⁴ It is the subject some debate whether one can still attack with a weapon that doesn't occupy the off-hand (e.g. spiked armor, unarmed strike). This is something about which you should ask the GM.
⁵ Whether the duelist's best abilities are also good abilities is another topic entirely.
⁶ While D&D 3.5 prohibited the feat Power Attack from being used with light weapons, Pathfinder doesn't. Even in Pathfinder, though, an off-hand weapon still doesn't reap the full benefits of the feat Power Attack.
⁷ Using oil of bless weapon in such a way means not adding to one's weapon any effects that trigger on a critical hit: "In addition, all critical hit rolls against evil foes are automatically successful, so every threat is a critical hit. This last effect does not apply to any weapon that already has a magical effect related to critical hits...."
⁸ Read that carefully. A character can totally be fun to play even if he only contributes narratively, and a character can be absolutely no fun to play while contributing massively mechanically. However, a character is often most fun to play when he can contribute both ways.

Based on the stats given, it appears to me that your best option is to not multiclass.

The most obvious option would be a dip into Monk, to get Unarmored Defense's Wis-to-AC bonus, but that would be trading leather's +1 for your Wisdom's +1. The other abilities wouldn't really add anything to what you already have.

Another option is Barbarian's Unarmored Defense. That would add +2 AC from your Con instead of your Leather armor, but simply changing to Studded Leather (also light, and giving full dex bonus) would achieve the same result. The Barbarian's Rage does add a moderately useful ability, but comes at the expense of delaying your progressing in Fighter/Monster Hunter.

Finally, a 2 level dip into ranger would allow you to pick a second fighting style, specifically "Defense" which grants +1 AC while wearing armor. Again, the Ranger's abilities do not synergize particularly with your other abilities, and delaying your main class abilities for 2 more levels is unlikely to be worth +1 AC.

Also note that none of these options combine. Unarmored Defense is specifically called out under multiclassing as an ability you can only get from a single class, and even the greatest Cheeselords still have trouble finding a way to be unarmored and wear armor at the same time.

It seems like you already have the best option figured out: take the Dual Wielder feat for the bonus AC, and indirect damage increase of 2 one-handed weapons.

While you're shopping for a second rapier, upgrade to a suit of studded leather. This and Dual Wielder will bring you up to 17 AC.

After that, just stick with fighter. At 7th level you can start using 2 superiority dice on damage, as well as automatically maxing superiority dice damage against certain types of enemies. At 10th level, your superiority dice get bigger by a step, becoming d10's.

These options will do more for you than you can get from any quick dips into other classes.

For a build like yours, and for most builds in 5e, multiclassing is a great way to increase the breadth of your skills and abilities, but usually does not add a lot to the depth of them.

As Ethan has pointed out in the comments, don't forget to turn to your teammates for assistance and buffs, as well as doing the same for them where you can.