In order to join the Sun covenant, you need to either have 50 faith, or you need to do a bunch of co-op. Each time you join someone's game as a phantom and help them kill a boss, the faith requirement drops by 5. So if you're not going to be putting points into faith, try to get summoned for co-op as much as you can at the early levels where there's still a lot of activity.
Make sure you collect as many Eyes of Death, Sun Medals, and Souvenirs of Reprisal as you can in each playthrough. You need 10 each.
Rhea's shop is easy to miss, and she sells miracles. After you
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rescue her in the Tomb of Giants, go back to Firelink and then take the elevator up into the Undead Parish to find her shop. If you visit too many areas before reaching her shop, Petrus will kill her.
Don't kill the Duke's Archive boss before rescuing Big Hat Logan and buying his spells.
After you max out a weapon and get the achievement for it, do not hit B to back out of the menu - you can turn your console off to avoid the autosave, and you'll still have your Slab to ascend another weapon.
If you have the Master Key, you can skip The Depths (including the boss), the sewers, and almost all of Blighttown. Pretty handy if you just want to plow through the game as quickly as possible (like in the third playthrough).
And this is totally optional, but it seems like you can only complete Solaire's questline on your first playthrough. I'm not 100% sure, but I think it's bugged and impossible to complete on all subsequent runs.
Probably some other stuff I'm forgetting, but that's all of the achievement-related things I can of off the top of my head.