Borderlands Lilith skill builds

I want to build a powerful damage dealer with good survivability but find myself very clueless as to what to put skill points in.

My Lilith has both of those characteristics, but it heavily relies on being able to find some good SMGs. Even still, she can't approach the levels of damage dealing that my Mordecai can dish out, but she is much tougher to kill due to her boosted survivability.

Here is how my skills break down with ~56 skill points:

Controller Tree

• Diva x5 (Survivability)
• Inner Glow x5 (more survivability when escaping)
• Hard to Get x5 (more phasewalking is good)
• Girl Power x5 (More survivability)
• Mind Games x1 (just for fun)

Elemental Tree

• Quick Silver x5 (More DPS)
• Spark x5 (Elemental effects are good)
• Resilience x5 (survivability)

Assassin Tree

• Slayer x5 (crit damage = more dps)
• Enforcer x5 (more DPS after a kill)
• High Velocity x5 (more accuracy and damage)
• Blackout x5 (more phasewalking, but get this after hard to get)

For the last remaining 14 skill points (or so depending ho what DLCs you have), you can think about maxing out Mind Games and putting some points into Intuition/Radiance. I wouldn't recommend phoenix though, as the flame aura from that can cause nearby barrels to explode, which is a pain in the ass on some maps.

With this build you can shoot and shoot, then run away with phasewalk when your health gets low.

I haven't tried a pure melee build as I don't think it'd work that well, but I could be wrong.

Finally, make absolutely sure you find yourself a class mod that provides SMG ammo regeneration, without it you'll go through ammo waaaaay too fast.

Preface: Mêlée, as a primary weapon, is negligible. This is a case of linear mêlée damage, quadratic bang bang. Although mêlée damage may not necessarily scale linearly, it's still bringing a knife to a gun-fight. At the upper levels, you should only use your mêlée attack offensively if you know the enemy's health is low enough that it will finish them off and you need to reload, or you want to Daze. More on Dazing later.

Skills:
There are three mêlée ones: Striking, Venom, Hit & Run, and Phase Strike.
Offensive: Spark, Slayer, Enforcer, Quicksilver, High Velocity, Radiance, Phoenix, and less so Mind Games, Intuition, and Dramatic Entrance. When I say less so, that is not to take away from the usefulness of the last three.
Defensive: Diva, Inner Glow, Silent Resolve, Girl Power, Resilience, Hard to Get, Blackout, and Hit & Run.

You should center skill builds around COMs, and I mean offensive COMs, like Specter, Mercenary, Tempest, Tormentor, Plaguebearer, and Firefly. Your build also differs based on the situation. The main situations include PvP, Underdome, PvE (story play-through), boss raiding.

By boss raiding, I pretty much mean Crawmerax the invincible. Crawmerax is only weak to critical hits. If you've got a sufficiently powerful Unforgiven Masher, the Specter COM allows you to one-shot each of Crawmerax's weak-points. Tempest is also good too because it also has Slayer (+crit) as a potential skill-boost, as well as helping to deal with Craw maggots. Radiance is great against Craw maggots. Radiance applies a long-lasting shock DoT, far longer than normal shock DoT, and this freezes Craw maggots in place.
If you're fighting Crawmerax alone such that the game pauses when you're in menu, it's useful to have a +4 Inner Glow Plaguebearer COM to heal up effectively every time you Phasewalk. You can use a Catalyst COM with up to +30% Team Cooldown Reduction to recharge Phasewalk more quickly.

In PvP, Radiance is your worst enemy because it gives away your position in Phasewalk. I don't think that Radiance even damages other players in PvP.
Hit & Run will allow you to scout more and play mind-games on your opponent. High Velocity (HV) can decide whether you hit first, and killing in one shot is common, so that's all the difference. Silent Resolve will prevent you from getting one-shotted. There are no critical hits in PvP, so Slayer is useless. There're not really any kills either, so on-kill skills are useless too. Pro-tip: On-kill skills (OKS) have a hexagonal outline.
Dazing is awesome in PvP, and while 5/5 Dramatic Entrance = 100% chance to Daze, sometimes it'll miss at point-blank. Striking actually becomes useful and clutch. So, PvP skill spec comes close to no question. The last two points go into either Venom or Phase Strike.
Shock damage is king because shield usually composes most of one's health (exception: Brick), so the Tempest COM is good, especially with +4 Silent Resolve, and shock artifact. In PvP, you should always be using a shock artifact. Fire is lowest tier element because it does the least damage against shields.
In contrast, fire is usually highest tier in other forms of combat, and Lilith has the best (and only) fire spec. out of all the other Borderlands characters, so I may speak in terms of Firefly.

Underdome: Top play in this combat form epitomizes the "powerful damage dealer with good survivability" that you seek. Although if you're really playing competitively, survival may be by the skin of your teeth, but that's the way to play if you want to earn the most kills and get pumped with the most adrenaline. Lilith has some of the greatest potential for this. Go from kill to kill, like a Diablo Barbarian. Blackout helps you keep up Silent Resolve. You're going to want to keep up Silent Resolve as much as possible, which means exiting Phasewalk quickly which detracts from Hit & Run. Keep up Girl Power and Phoenix. Intuition is not really for survival but to get to your next kill. You'll blaze a trail and really be on fire. And remember that good defense is great offense. More killing = fewer enemies.

Caveat: You can't necessarily play a style if you don't have the good enough gear.

Eridian artifact: While there's an obvious synergy of fire with Firefly and my testing shows that same-element DoTs can stack, you'll already be shooting fire. Shock and fire have visually disruptive explosions. I want to say that explosive does less damage overall than corrosive. Here's some quick testing that I did on skags:
"984, 611 explosive Phasewalk. 1053, 657. Corrosive: 66-69, once every 1/3 seconds or so. Ended with 28. It lasts for 22.4 seconds on Alpha Skag. On Alpha Skag, it dealt from 80 to about 180 damage, varying. On another test, it lasted 15.2 seconds. Initial damage is higher, in the 3-4 hundreds. Before Alpha Skag, it was little 'uns, like colicky."
So, I'm gonna say that corrosive does more damage than explosive. And it's very satisfying to see all elemental DoTs on a baddy, via Hellfire, Radiance, and corrosive artifact.