I play a 4e Drow Assassin ( Night Stalker ). I find that the class while a striker does not do much damage.
Would it be better if I play a hybrid or keep to pure Assassin.
I don't think that the class is outright broken (as in "unplayable") but rather is seriously lacking feat/item support and requires a much higher level of expertise to properly build and play than other strikes (especially the ranger, where spamming Twin Strike is almost always the perfect answer to any problem -_-').
Anyway, there are some ways to increase the damage output of an assassin. The primary thing to remember, however, is that you will primarily deal large spikes of damage and not a high baseline damage every round (like e.g. a rogue).
Also note that an assassin can basically use the same tricks as other characters as well to pile up damage (using BFG-style weapons like the executioner's axe or the fullblade, knocking prone for Headsman's Chop[DDI], using a Frost Weapon[DDI] for frost cheese, using items like Gauntlets of Blood[DDI] for more damage against bloodied targets, and so on and so forth).
The original assassin is severely underpowered in typical scenarios.
To raise an exception to your wording, the assassin is not "broken" in the normal sense of the word (game-breakingly over-powered) but is broken in the "lying on pieces on the floor" sense. It does not compete in damage, though its mobility options are exceptional. It is, however, worth noting that a strong stealth empahsis on your party can make assassins quite effective. Unfortunately, in typical tactics an Assassin doing its thing is the equivalent of a hacker in SR3 shadowrun: while they do their thing, the rest of the group goes and gets pizza.
My recommendation is to call a different class an assassin. A shade (race) or rogue cunning stalker will give necessary "assassiny" stealth while allowing non-zero damage. An essentials drow thief can be quite exceptional in combat, considering the damage output of drow feats on crossbows. If you provide your build, I will attempt to recreate its playstyle in a class that doesn't suck. (In fact, I've already statted a base drow "assassin" here)
If you're interested in a hybrid, I recommend this guide: "The Eladrin Eldritch Executioner and its Variants" It links to a few interesting builds.
I've had no problems producing Striker Damage with my original assassin.
As a Drow you should comfortably have 18 Dex 18 Cha.
The 3 feats I think are most important to my character's success are; Sneak of Shadows (you can reliably get CA as an Assassin giving you a 1/Enc +2d6), Killer Instinct (gets that 2nd Shroud down round 1), and Light Blade Expertise (again easy to get CA).
With Shadow Storm you should be getting 1d8+4+4+1 every round as a base, get a flank with a buddy and you have 1d8+4+4+2+1. Never build more than 2 Shrouds except if you used Killer Instinct.
For Theme the Unseelie Agent is an excellent choice for Assassins; it gives you a summonable +1 weapon at level 1. (OK its great for everyone but the whole "I'm unarmed but can summon a weapon when I want it" thing thematically fits the Assassin.) This will propel you to 1d8+4+4+1+1 as a base damage.
Finally the Death Shroud Ki Focus is the "go to" Ki Focus for the the Assassin. It lets you put an extra shroud on a creature after you have hit it - with your Shrouds already declared as being used.
This build lets you do a turn 1 spike damage (if you can get CA) of 2[w]+4+4+1+1+2d6+3d6. That's pretty safe striker damage. Sure you drop down to 1[w]+10 as your base after that but you should be able to routinely get an extra 2d6+1 damage on that.
If you take Two_weapon fighting (and get a Rhythm Blade for your off-hand) you get another +1 damage.
You can also access Nimble Blade for an extra +1 to hit making it more likely you will add your Shrouds more likely to be triggered.