Pix has 5 abilities, Glitterlance is a high impact, short duration slow; Whimsy grants movement speed or a polymorph; Wild Growth increases health, size and knocks the opponents in the air while also slowing them; and Help, Pix! is a shield or damage ability. These are all really useful utility spells and seem to point in the direction of support, indeed they do. But... would Lulu agree? I think not!
Lulu's passive Pix, Faerie Companion deals damage each time Lulu attacks (or the friendly target he is on. This doesn't really feel too support-like. Most of the time this seems useless, unless it is on the AD-Carry, who will probably attack faster than we do.

But wouldn't it be profitable if we could make more use of that passive? We would be able to do so if it gains more attacks. Which means we attack more, it will be more useful.
But if we want to attack more, we need to have attack speed, which is not a support stat, also damage is not the main priority of a support. I guess our passive is not so good...

But what if we DO go for damage? Our passive would increase in potential. But what of our other abilities? Let's review them: Glitterlance, would not benefit from it, is still useful and does not lose anything from it; Whimsy does not particularly benefit from it, is still useful, it even increases in potential to use because we can chase or polymorph enemies we attack; Help, Pix! was a damage skill, which is good to us when we want to do damage, so it will gain us some potential; Wild Growth will not benefit from it, but because of it's cool crowd control mechanics, be useful to us.

So that means our skills do not lose potential because of us going for damage dealing. Which means we can be a damage dealer providing some awesome utility skills to the team.

But, if we do go as a damage dealer using attack speed, where will we play. Well we got Glitterlance which got a short duration, but slows our enemy for quite a bit; and we got Whimsy which makes us move faster.

We take Madred's Bloodrazor because it pretty much provides the exact stats we long for, attack speed and a neat on hit effect that gives an edge against tanks especially. It also provides some attack damage to increase our auto attacks a bit.

Normally Madred's Bloodrazor would be a situational item, but we use an on-hit playstyle, combining auto attack damage with magic damage from procs that trigger from auto attacks. This will play out much like a Teemo. We pick Madred's Bloodrazor because, for the on-hit playstyle, it is actually the strongest item. It is not situational in the on-hit playstyle, it is must have even.

This item will help a lot with our ganking, that's the reason we get a part ( Phage) pretty early in the game. It will slow our enemies, which means we can auto attack more while they are running. We upgrade to Frozen Mallet later to tackle our squishyness. An alternative to this item is Trinity Force which provides the same on hit effect, a bit of attack speed and some more general stats. But, as stated above, it will not really do too much against our squishyness, it will give us some more damage output however.

Vampiric Scepter as an opening does not suit us really well because of the way our damage is inflicted. Especially at the start of the game, half of our damage is magic damage (the on-hit effect of Pix, Faerie Companion deal magic damage), which means the health that we would get from attacking is fairly low. The Health Potions from the Cloth Armor opening provide us with way more health early game.

Also, you'll notice later on that I tend to go for life steal quite late. That is mostly because it will not benefit our damage. I think faster jungle clearing gives us more ganking potential and therefore that is the way to go. Cloth Armor does not give us any higher damage, but the upgraded version Madred's Razors will (especially against mobs.)

Regrowth Pendant will grant some long term survivability, but doesn't protect us during our initial fights. On top of that we don't really use this item later on, we don't get the Philosopher's Stone because we have another way to get the gold, namely ganks.

This item will compromise our survivability, but will enable some better ganks (which in my opinion are sufficient with Cloth Armor.) Also, we will not use in the late game.

The boots would compromise our survivability by quite a bit. If you want to do Blue, Red, 1 or 2 gank attempts and all in between (except Golems) you'll need the sustain given by the recommended opening.