Bloodthirster

Why get it on any character over Infinity Edge or Mandred's Bloodrazor? Do some champions synergize better with it than others? Who would prefer it and why?

If you do decide to choose Bloodthirster over the other uber weapons, should it be rushed? It is tempting to do so since the Vampiric Scepter start is a common one. Is there a hurry to build up those stacks or are you better off waiting until you have a few smaller items before investing in one so expensive and undependable?

Well 25% life steal is a pretty good combo with +100 damage. As long as you are staying alive for at least a little while it is pretty easy to farm 20+ minion kills in a minute or two, making it the highest damage item.

Also if you are playing a character that doesn't really need crit chance or IE's passive, you only need 3 minion kills for Bloodthirster to have a better damage/cost ratio. If you fill it up it has a much better ratio than IE, and also a better ratio then Black Cleaver.

I get bloodthirster(s) on both Pantheon and Gangplank (after update). Both these champions have spells that are very dependent on attack damage.
For pantheon I generally rush a bloodthirster (start with boots->bf sword-> bloodthirster) This gives a needed boost to damage, and the life steal makes it incredibly easy to kill golem/lizard/dragon generally either gaining health or staying even. I sometimes get a brutalizer, then bloodthirster if I had to go back early, as the cd reduction from brutalizer helps with your long cd skills (relatively).

I get bloodthirster on Gangplank after IE and PD. This is when I already have 70% crit, so getting more is basically a waste. I get bloodthirster cause a full one would add 110 base damage to Parrrley, or about 300 on a crit. Also, since Parrrley can life steal, it is very nice to get back 300 health per parrrley with 25% life steal.

Basically its good on champions that need life steal and damage, and either already have IE or don't really need it.

Never get it. UNLESS your champ scales off Damage and you'd benefit more from the lifesteal than getting a Stark's / Exec.

It provides the single largest attack damage boost of any item in the game, as well as the single largest Lifesteal of any item in the game. The former is very good on champions that have abilities that scale off of base damage (Katarina's Bouncing Blade and Death Lotus, Pantheon's Spear and Heartstabber, Corki's Gatling Gun, Shaco's Shiv, Tryndamere's Spin, Ashe's Volley, Gangplank's Parrrley, though those last four benefit more from Infinity Edge as they benefit greatly from crit chance/damage). For Kat especially, Bloodthirsters are vital. However, it's generally better to acquire a couple of BF Swords before upgrading to Bloodthirsters. Madred's is generally a lesser case on most DPS heroes, and should really only be built if your enemies are stacking a lot of HP (or if you are facing down Cho/Mundo). Last Whisper generally gives a lot more cost effective damage output for a +40% attack speed item.
Once you have a Bloodthirster, it's not too hard to build up stacks. Your damage is generally high enough to clean up waves of minions, particularly if you're using a splash ability like KI'd Bouncing Blade, Heartstabber, or Gatling.

Thanks for the excellent responses. What I gather is that unless ur playing Katarina, Pantheon or Corki, there is no point in considering getting a Bloodthirster at all, much less rushing one. Thanks for the advice.

I usually go Bloodthirster first for a few reasons.

1). It provides early game HP regen for a low cost. (Vampiric Scythe)

2.) It provides leverage/survivability 1v1.

3.) It's faster to get than Infinity Edge and in my experience, uses more immediately
useful items in its construction. Both use BF sword, but the Bloodthirster adds that lifesteal
that I need to either lane indefinitely or to go 1v1 against another champion. The Infinity Edge uses the Pick Axe for damage (which is immediately useful) and the crit cloak (nominal unless you've been stacking crit the entire time).

Basically it boils down to, I prefer the lifesteal over the +25 damage the pickaxe provides.

(I play Ashe almost exclusively).

Cleaver will do you better good than Bloodthirster on Corki.