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).
7[W]but also let's you use Assassin's Shroud up to 4 times against the target after the attack; useful for building chains:
add 4 shrouds in any way > attack with this power > immediately have 4 shrouds again > spend action point and attack with other daily/encounter
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.