Ah, Philosopher's Stone. Consistently ranked second to last of the cards with Potion in their cost. Why are you so unloved?
The answer is pretty simple, actually. Philosopher's Stone needs you to not draw your deck, and drawing your deck is awesome. Drawing your deck is so strong that you will pretty much never opt to pursue Philosopher's Stone instead, given the choice. So, at minimum, you should only consider P.Stone when drawing your deck is either impossible or really impractical. For the rest of this post, let's take it as given that you can't draw your deck.
Even then, Philosopher's Stone is a tough sell without other Alchemy cards available. The kind of deck that wants Philosopher's Stone can't buy it very often because it takes several hands to make it through each shuffle, and you can buy at most one P.Stone per Potion per shuffle. And it's a risk because you may not have $3 in your hand with your Potion(s). And when do you buy your first Potion? Early on, Philosopher's Stone is worth $1 or $2. But if you wait too long, you won't be able to accumulate enough of them.
When do you go for Philosopher's Stone, then? Certain other Potion-cost cards make it a no-brainer. Familiar is chief among them. You deck is filling with Curses and you only want a couple of Familiars. After that, all your Potion buys should be P.Stones. University is another good one. Usually you only want so many of those for gaining $5 Action cards (remember, we're assuming that you can't draw your deck). Even Transmute can be enough, assuming you don't have another great way to trash your Estates. First buy the Transmute, then P.Stones.
Anything that allows you to play your Philosopher's Stones more often is good, especially if it doesn't lower P.Stone's value. Herbalist is the big one; it gives you +1 Buy and allows you to topdeck a played P.Stone. Warehouse and Cellar are nice. Cartographer removes a card from your deck/discard, but is still a good option. Scavenger is great.
Gaining cards is good, especially if the gainer gets you a benefit for your current turn. Think Beggar and Explorer.
When you are playing a Philosopher's Stone deck, buy Duchies and other cheap Victory cards much earlier than you normally would. Usually they would slow your deck down and prevent you from buying Provinces, but the P.Stones will let you spike $8 hands anyway and you may need those extra points to overcome more conventional decks.