Pretty much everything's already been said with all the really good advice in the posts above, but if I may add my two cents, I would add that, choosing a load out that privileges high burst damage rather than constant damage over time is probably a good idea, as it tends to encourage a different play style from the typical face tanking that very often results from the complacency of having near unbreakable shields.
For example, I run a fairly typical hybrid setup on my FAS, with a paper thin bi-weave screwed on top of a heap of HRPs and MRPs, with a full set of double-shot frag cannons and it works very well, in addition to be of total blast to play (
click here for the full config)
Because the frag cannons require to almost shoot point blank in order to be of any use, you tend to naturally dive on the target, boosting at full speed while closing in and deliver the shot right before you pass it by (loudly shouting "Yeeehaaaa!!!" as you do it optional but highly recommended).
Just remember to *never* fly in a straight line and use vertical /lateral thrusters while correcting your trajectory with pitch/yaw, as that'll render you almost immune to slow moving projectile weapons at anything but very close range (at least from NPC). Meanwhile your shields, as thin as they maybe, will soak up most damages from lasers, as long as you've engineered some thermal resistance on them, because you won't spend so much time right in your enemies' line of fire.
And because frags are so easy on the PD and shields are only here as a buffer, you can fly in a 1/4/1 pip configuration almost constantly, making the strategy described above even more effective: a FAS with C5 DD in perma-boost is a fast beast indeed.
Of course you're not going to spend hours on end in a RES with such a setup, as you'll burn through ammo real fast (but again, it does kinda coral you into being more disciplined, making every shot count rather that firing willy nilly all the time...), but survivability in PVE has never become even close to being an issue for me with this outfit.
Admittedly, this is an hybrid build and the shields are up most of the time - they go down fast, but the also get back up fast - which makes it easier on your modules than a pure hull tank, but although I have yet to try it, I'm pretty sure using an AFMU and chaff instead as someone suggested would work equally well; the trick being to avoid locking yourself up in a deadly embrace with your adversary, taking nearly as much damage as you dish out.
