Do this for skills.

Take aim is your main priority, it opens the gap between you and the bad guys, making you harder to kill, additionally above 3 levels your auto attacks are outside tower range.
Headshot adds some nice damage, but it's no good being able to hurt things if you're dead.
Assassinate is what makes sniper very, very useful, it's long range and high damage, makes you very, very effective at dispatching enemy heroes who are running away, second only to Zeus from what I've seen.
Shrapnel is pretty crappy until very end game, so just ignore it until you're making your final push, then stick a point in there to give you another source of income.
Items wise, go for:
Boots of speed
Crystalis
Upgrade boots to Power Treads
Upgrade Crystalis to Daedalus
Buy another Crystalis (The buffs are seperate, most attacks will get the crit modifier)
If you've not won by this point...
Shadow blade
Mask of madness
Monkey king bar
You can pick up the mask earlier if you're having trouble with towers (the attack speed clicky makes it easier to get them down).
Admittedly, this is a high cost path for gear, relies pretty much on you not sucking and killing lots, but also being pushed enough by the enemy that the game lasts a while.
If it's a bit easy, or you're not making a lot of cash go:
Boots of speed
Mask of madness
Upgrade boots to Power Treads
Crystalis
Monkey king bar