Which assassins or melee champions have the most significant CC?

There is no real way to determine which CC is the best, because they all depend on several factors. Some are AoE, others are point and click, others are cages, some don't even do damage. I will try to mention the best in each category, for the sake of understanding

Stuns/Roots

Duration - The stun that lasts the longer is Ashe's Enchanted Crystal Arrow, which can last from 1 second, up to 3.5 seconds depending on distance traveled. Irelia's Equilibrium Strike, which lasts 2 seconds at max rank, but requires the user to be with less health than the target. It's tied with Brand's Sear, except Brand's Sear lasts 2 seconds at every rank, however, to stun, the target needs to be ablaze first. Elise's Cocoon lasts 2 seconds at max rank, but is a skillshot. There's also Veigar's Event Horizon, which requires the enemy to touch the cage wall to stun, for a maximum of 2.5 seconds. Then, there's Morgana's Dark Binding which can root for up to 3 seconds.

Damage - This one is hard to factor. If we consider single target damage, AoE stuns won't be very effeective, however on teamfights,they are considerably stronger. Then there's the issue with scaling (Dark Binding does 80+95%AP at level 1, versus Equilibrium Strike which does 80+50%AP. However if Irelia has sufficient AP, she deals more damage). There is also the issue with builds, an AP Morgana functions differently than a Support Morgana, and as such the damage provided by the root will differ. Secondly, usefullness can also be a factor. A full AP Amumu can do a LOT of damage with his ultimate, however full AP Amumu is extremely weak against organized teams. As such, it's not possible to take damage into account.

Special Mentions - You have Annie's passive, which stuns every 4th damaging ability. This is interesting because it can be a point and click projectile (Q), a AoE instant Cone (W) or a target area (R), while the duration and damage is the same, it's much stronger to use R with the passive (due to being instant, and unavoideable - if well executed) than using it with her Q or W. Braum's Passive requires his team to attack the target, meaning, while the duration is the same, where the target is stunned depends on how both teams move and attack, which can mean less reliability. It can also mean more or less damage due to it's second effect, which makes the target take extra damage from Braum for a few seconds. Depending on Braum's attack speed, the total damage inflicted by the stun will vary)

In General:

League of Legends doesn't offer good superlatives. While one stun can do more damage, maybe it's so hard to execute that it's weaker. Maybe one stun can last an eternity, but depends on other factors (eg: Braum's Passive, Ashe's R). Or maybe it requires you to be at a disadvantage (Irelia's E). Then you have to consider who it's targeting. Taric can't miss his stun, but it only hits 1 person and it takes a while to affect him. on the other hand, Ryze's W is instant, but lasts less time. Then you have Amumu's R which is instant, and AoE, but you can still be attacked (eg: doesn't stop katarina's ult). You even have Galio's R which does more damage the higher the enemy attack speed and position and range the enemy has, or even Tahm Kench's W, which can move the enemy around.

Post Original Post Edit:

Ekko is the assassin with the strongest CC. He can stun all enemies that are in his W zone, if he's also in it. It can stun up to as many enemies as are in the zone, while giving a good AP scaling shield on top of it.

Champions in the Assassin Category Who are Melee and Have Hard CC

^{--All numbers are from skill rank 5 and champion level 18 so durations and damage will certainly vary--}

• Ekko
  • Passive: 60% Slow for 3s, 190 + .8 * AP magic damage
  • Q: 60% Slow, 280 + .7 * AP magic damage (if you hit both directions)
  • W: AoE Stun for 2.25s, no damage
• Fiora
  • W: 50% AS/MS Slow for 1.5s (stun same duration if parrying hard cc), 250 + AP magic damage
  • E: 60% Slow for 1s, damage equal to one auto and one critical auto
• Fizz
  • E: 60% Slow for 2s, 270 + .75 * AP magic damage
  • R: AoE 70% Slow for 1.5s, Knock-Up for 1s, 450 + AP magic damage
• Irelia
  • E: 60% Slow OR stun for 2s, 240 + .5 * AP magic damage
• Jax
  • E: AoE stun for 1s, (150 + .5 * bAD) * 2 (maximum, if 5 autos are taken during wind-up) physical damage
• Nocturne
  • E: Flee for 2s, 260 + AP magic damage
• Pantheon
  • W: Stun for 1s, 150 + AP magic damage
  • R: AoE 35% Slow for 1s, (up to) 1000 + .5 * AP magic damage
• Poppy
  • Q: AoE 40% Slow for (up to) 1s, (160 + .65 * bAD + 6% target max HP) * 2 (if both procs land)
  • W: Enemies who dash in the field are knocked-down, 230 + .7 * AP magic damage
  • E: If rammed into a wall stun for 2s, (130 + .5 * bAD) * 2 physical damage
  • R: Knock-back, 400 + .9 * bAD physical damage
• Rengar
  • E: 80% Slow for 2.5s, 250 + .7 * bAD physical damage OR root for 1.75s, 590 + .7 * bAD physical damage
• Riven
  • Q: 3rd proc AoE Knock-up, 90 + .6 * AD physical damage
  • W: AoE stun for .75s, 170 + bAD physical damage
• Shaco
  • W: AoE Flee for 1.5s, 95 + .2 * AP magic damage per attack
  • E: On auto - 30% slow for 2s, AD physical damage | On cast - 30% slow for 3s, 210 + AP + bAD magic damage
• Vi
  • Q: Knock-back, 240 + 1.2 * bAD physical damage
  • R: Knock-up for 1.25s, 450 + 1.4 * bAD physical damage
• Yasuo
  • Q: On third use, AoE Knock-up, 100 + AD physical damage
  • R: Suspend for 1s, 400 + 1.5 * bAD physical damage

Analysis

Looking at the champions above, you can see the the longest lasting CC comes from Ekko, but only by .25s. It also requires significant planning ahead on the part of the player to land this one.

The most damaging is probably Fizz's ult, which also does you the favor of increasing all magic damage done to your target by 20%.

In my opinion, Ekko would probably fit best into your requirements. However, don't just take my word. Try out lots of different champions. There are a lot of champions without hard CC that you may enjoy more than those listed above.

Glossary

AP - Ability Power
AD - Total Attack Damage
bAD - Bonus Attack Damage
AS - Attack Speed
MS - Move Speed
AoE - Area of Effect (size and shape of area always vary)
Flee - Enemy cannot be controlled and runs away from source of flee
Stun - Enemy cannot take most item, skill, or summoner spell actions
Knocked-Up - Enemy is knocked into the air briefly and cannot take actions. Cannot be cleansed.
Knocked back - Enemy is knocked away from source and cannot take actions. Cannot be cleansed.
Knocked down - See Knocked-Up.

^{Order of operations apply to all math (e.g. multiplication before addition, etc.)}

This question is extremely hard to answer since it is asked in a extremely vague way, but I'll try to answer them as good as possible

Highest Damage

If you ask for ANY cc on a champ at lvl 18 without items, the highest damaging CC Ability would probably be Wukongs Ultimate with a 200 (+120% AD) scaling, which is active for 4 seconds. Wukong has 60 AD at lvl 1, with additional 3.2 AD per lvl. This equals:

60 + 3.2 * 18 = 117.6 AD

So in the end, the ultimate deals:

(200 + (117.8 / 100 * 120)) * 4 = 1365.44 

This would even surpass Nunus Ultimate, with a base dmg of 1125 at rank 3.

Strongest CC

If you ask for the skill which deals a lot of damage and brings a lot of CC, its most likely Galios Ultimate, which can deal 720 + 108% AP to every enemy if galio gets the entire amplificiation, which he easiely gets in a 5 man ult.

Longest Stun

The longest real stun in the game is Veigar with 2.5 seconds on max rank.

The longest unreal stun is Morganas Q, with a stupid high duration for 3 seconds.

I haven't seen one of the CC's that is often used in Wombo Combos: knockups, knockbacks or any displacement effect. It's nice being able to stun somebody for 2.5 seconds, slow them by 95% or so for 4 seconds (Nasus' Wither) or root them for 3 seconds. If the enemy has cleanse, Mercurial Scimitar, Mikeal's Crucible or oranges, the duration is meaningless.

And that's where knockups and knockbacks come in. They cannot be removed by any cleansing effects and tenacity has no effect on it. Meaning you can actually stop an Irelia slicing through your team. You will be disabled when hit by knockups/knockbacks which in my opinion makes them superior to other kinds of CC.

Their base damage is often not that high, but those abilities tend to have good ratios. Malphites ultimate Unstopable Force for instance, has a 100% AP ratio. Slighlty ironic that a Bard or Sion actually stop it though..

Also note, Nami's bubble and Yasou's ultimate are not knockups and they can be cleansed.

If you allow gimmicks build (since you're looking for the most damage AOE and the longest stun) it is quite easy to answer your two questions (Oak did pretty well to be honnest).

Since Oak didn't mention slows a lot i'm going to talk a bit about these tilting machines.

The biggest CC aoe that deals damage is nunu, since the fully channeled ultimate is the biggest dealing damage spell in the game AND does CC (a slow).

The strongest single target slow is Nasus, since it is a point and click spell, last longer and slow more than almost every other slow.
Honorable mention for thresh that slows at 99% with his cage in aoe.

i'd like to add some other CCs not mentionned, which are taunts or fear. Rammus taunt and fiddlestick fear are both very long cc, that lasts 2.25 second on max rank.

Bonus : The potential chained stun/root max duration by one char used to be ryze, who could cage you during 4.2 seconds alone (not sure if still possible)