Some basics:
Pick up two points in armor before both of your other skills are maxed. Usually take armor first at level 2, then the next point earlier if you're getting harassed a lot, sometimes at 4. Make sure you get two points before mid game and be mindful of your armor going in to fights.
Keep armor stacks up in lane by clicking on heroes and letting creeps get some swings in, this is very important.
Try to remember which trees you have destroyed in the short term, escaping from that second gank attempt can be hard if you already destroyed your best lifeline.
Remember that you can use skills together. Ex: Use your Q during timber chain to hit multiple targets, hitting a tree before you start moving for maximum effect. Use Q or chakram to clear trees ahead of you to timber chain to different positions.
The key to playing timber well is knowing exactly how much damage you can do while managing your mana, kind of like storm. Go in, get your damage in, but make sure you have a way out. You need to be aggressive once you get your full kit of skills, there are very few heroes that can keep up with you in damage output. Be very, very careful of getting silenced/disabled and picked off, you're going to want to dive and go nuts, don't do it.
Items: Don't write off soul ring early. I usually skip it if I'm playing against a tri lane, but 1v1 it's really, really good. Other items to consider are atos, blink, and ags, ags especially if you're playing against a team that wants to push really hard early. Going uphill against two chakrams is brutal, and makes the smaller time you get to farm more efficient. After a lot of messing around with item builds I think bloodstone take precedence over anything else and is core on the hero, so keep that up.
With regard to bkb, only get it when you absolutely need to. I think timber is a hero that really can take advantage of a heart and some magic armor and soak up a lot of damage for your team. Chain away, regen, come back in.
Hm, just realized I probably paid too little attention to armor stacks, I'll try to keep an eye on that in the future. Between chain and whirling death, what should I prioritize? I guess chain because it doubles as an escape/chasing mechanism? For fights, I usually try to chain through heroes and use whirling death when near them. What do you mean with 'hitting a tree before moving for maximum effect'? Does it do anything for the chain, or is it only for the pure damage of whirling death? What about shiva's and hex? I always got shiva's for the extra tankiness and the aoe effect, but I'm thinking my games would probably have gone better if I'd gotten a scepter instead.