There are different ways to play Storm. Following is the way I play him and I'm not going to go into a detailed defense or comparison with other ways to play him.
As a mid hero, Storm requires quite a bit of resources to come online. Compared with Night Stalker, or Puck, or even Queen of Pain, who are often expected to open up the game for their teams, Storm often needs his team to open up the game for him! (Just recently we saw iG.Banana do nothing in the early game but stack the jungle for Ferrari). His greatest strength is in picking off split pushers and he is a hard counter to Tinker and Furion. A good strategy is to run split push yourself to force a split game where Storm can shine. Pushing also takes pressure off Storm in the early game tempo. Against team push games, he can be quite impotent even when faced with few disables.
Starting Items
I've experimented with different starting builds. I've used straight bottle, reasoning that we start with a spam-mable, high damage AoE. I've used a more standard tango 3 branch start.
My favorite in the end is tango, pot, double mantle, double branch. (in the new patch you will probably have to mantle + 3 branches)
Item Progression
Bottle, Null Talisman (Wand), Treads, Orchid
From here you can adjust as situations permit. To split push and become harder to kill, you can Linken's. To fight against heavy disabling teams, BKB is an option. To maximize your team contribution, continue Hex and Shiva's.
Skill Build
QEEWER WWEW, etc.
N.B. this is no longer the standard build. People usually 1-3-1 now, but I still 1-1-3 then put points into W.
Laning
Storm's Q at level 1 has very high damage and makes cs-ing quite reliable. Once you have level 2 with E, you have a high damage physical attack too. You can either use both Q and E to cs or use Q to cs and then deny and harass.
Be aggressive and harass the opponent with E charges. Your goal in the early game is to get fat and have as early an Orchid as possible.
General Play
E is your main source of damage and this is why I put early points in it (and this used to be standard). Before level 6, if you are trying to kill someone with a rune for example, use the low cooldown Q to charge up E before you go in. (always consider how you are going to have a charge of E on your very first attack. Specifically, if you are opening with an invis. rune you should Q before going invis. even if you are level 6. use Q often for aerial vision as well).
After level 6, you should use your ult to "roll" and recharge between attacks by just ult-ing in the same spot.
To combo once you have 6, you Ult in, attack to use the E charge, pull and use Q, attack, and keep rolling to charge between attacks.
Now Storm's Ultimate has some special details. 1) you are invulnerable. This means you can just Ult out of Puck's ultimate, dodge undodgable abilities (similar to Manta's brief invulnerability), and disjoint things such as Tinker's rockets.
You can also use items while flying. You can Orchid a target before you arrive, start channeling TP before you've landed, etc.
Playing Storm you must have good mana management! Always be aware of your mana status and have an escape plan. Learn to cut treads when you cast (but not in fights when you may be stunned).
Know the limits of your mana pool! As a rule of thumb, I prefer to limit my jumping to blink range. This increases landing accuracy, and more importantly, YOU WILL NOT RUN OUT OF MANA right after killing someone. Learn to come out of fog and let teammates lead. Learn to use E charges for damage and not ult-ing distance. At the extreme, it is in my opinion a sign of a poor player who makes a habit of taking jumps distances larger than about 1800.
Exceptions are of course made for when you have a regeneration rune buff or the fountain buff. Remember that after teleporting from the fountain, you retain the buff for awhile and can fly a bit further and even use a bottle charge. But you should still refrain from jumping more than 1800 units! You will just have to walk right back to the well.
Fun hero, specific strengths, good luck.
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