First of all lets find a denominator for an assassine, the role or job if you want to call it. Like with clerics, which ahve the job to heal, and warriors which have the job to take incoming dmg and dish out some dmg, there has to be role for rogues. But since we are not interested(and rift offers such a huge diversity in roles) in all rogue souls, lets just focus on assassines.
To make a long story short: The job of the assassine is to assassinate someone! I even would go that far to say: Noone cares if the assassine dies doing the job. As long as its target is dead, the assassine has done its job!

Why cant the Assassine do the job right now? Well there are several problems which i will elaborate further:

1. Positioning
Usually PvP Battles are about positioning, and Rift is no exception. THe usual Positioning is
range->melee<->enemy melee<-enemy range
The Main target of the assassine is the enemy cleric which usually hides in the enemy range position. And since your full damage capacity is reached when you are behind your target you are at the end of the field. So in the end the field looks like this:
your healer->your melee <-> enemy melee<- enemy healer <- Assassine
This positioning makes it impossible for your healer to give you any heals! (Tahts the reason i end up with 800 healing taken in so many warfronts). This alone would not be the problem. Its your job. Go there kill the enemy cleric.

2. Walking away
The best way to avoid assassines is, jokingly, to just walk away. Your reaction time+(if you walk away the assassine has to react to your walk away, so you gain distance and ping adds to this (or removes) a little bit) is JUST enough to be able to just walk away from an assassine while keep spaming instant heals.(more on this later)

3. Global Cooldowns
The WORST enemy of any assassine right now is not bad healing or low damage, its global cooldowns! Believe it or not, usually a pvp soul equipped assassine has ALOT of stuff against a healer. But the global cooldowns work so much in favour for the healer its a shame they were implemented that way.
Usually an Assassine should do the following: approach in stealth the enemy and dish out max dmg and so gain an advantage the enemy healer cannot undo and dies.
But in reality he cant do it! Here are the things an Assassine has to do in order to HOPEFULLY be able to kill a healer:
Come out of stealth with alot of dmg or with a stun. If enemy healer has PvP soul, stun is the wrong decision -> you just gave up your advantage!
Now, NO MATTER WHAT, the assassin HAS to use a snare. If not the enemy just walks away. If the enemy is clever he jsut uses dispel and walks away ANYWAY! -> you just lost your fight!
If he has the PvP Soul(which i assume every Assassine has in a Warfront) he has two more GLobal cooldowns to use: One to purge one buff, one of the zigtrillion buffs all players got, and hope he gets the right one. And another to reduce the healing of the enemy Healer, becasue he WILL heal. There is right now no (well equipped) cleric that will die in a stunlock!(There usually is so much aoe heal and he has so much buffs/hots7armor its just not doable on a pvp equipped healer)
So there are 3 global cooldown WASTED to do something just to be able to kill the cleric and not actually killing the cleric! ANd please remember, if he dispels your snare, he just walks away from you!

4. Low Damage
Comapred to a mage Assassines do low dmg. Compared to a pyro assassines even have less, range, less CC, less mobility AND less dmg. PYros are the better Assassines!

So, how can we change this without completely trainwrecking clerics? Here are some suggestions:

1. Add more poisons and a skill that lets Assassines use more than one poison for each weapon! (if i poison my blade with 3 poisons, 2 of them wont be gone, the blade is jsut dripping in poison! Its no magic enchantment! Its poison!)
What poisons should be added:
1.1 - Take the (if i remember correctly) 44 Point of Necros and make a poison out of it: Healing on the poisoned target will do X amount of dmg per stack the poison is on the enemy. The poison stacks up to Y times! The longer you are poisoned the higher the chance you die!
1.2 - Move = Dmg Poison: If you are poisoned and walk you take X amount of dmg for every stack of poison every step you take!
The point of this (pretty strong) new pisons is simple: If you keep on healing and keep on ignoring the assassine(as it is done right now) you die! You HAVE TO focus on the attack and defend yourself(dispel) against the assassine in order to survive. YOu cant just keep AOE heal spaming and survive and heal around you too! There is no way simple aoe heals should make assassines useless!

2. Add a skill that attacking from stealth will add one stack of poison from each poison with 100% chance.

3. Successfully killing an enemy will give you a 2 second buff that allows you to enter a stealth mode. If you enter stealth in that time you are in a weakly stealth mode. Easy detectable(from far away) for X amount of seconds! This allows for you to actually play again as an assassine even if you are in midfight but dont go into godmode every time you kill someone. You just killed a cleric, everyone saw you, they still can see you, but they have to decide: attack the assassine so he doesnt get his advantage from stealth or keep on attacking the current enemy or run.

4. Make the PvP debuffs(-healing and purge) castable during stealth without breaking it!

Thats about everything an assassine would need right now.
On the other side you could just increase the dmg of the assassine and giive them 5k crits on his enemys but somehow i think that would just make it dull to play one.
I am fully aware that i did not give any suggestions on how to solve the positioning problem. I didn give any suggestions because i dont see it as a problem for the assassine. I see it as the downside to play an assassine. You are alone you are behind enemy lines. you are on yourself. go do your job noone cares if you die!
I am as well fully aware that the PvP debuff(-healing) and the new poison(dmg when healed) COULD be devastating for any healing class ... until he finds a debuff button. Debuffs take more debuffs away than one with each cast. So i think it should be fine. ANd the poison gets worse the more stacks are up while the healing debuff actually decreases the more the target is healed. SO when the poison is at max stacks, the -healing debuff should already be gone.