Getting better as carry Natures Prophet

Start with a Clarity or two, a set of Tangos, and some Iron Branches, and head to lane. Make sure you use your nice, long attack range and good base stats to harass and last hit safely. If the enemy pulls, spawn Nature's Call and send them to cut off the creep wave and bring them back to you, to last hit safely. Next, get your Boots. I usually get phase since they're great for early agression but get treads if you really need the extra durability. You mentioned buying Ring of Basilius, which is okay, but generally you'd rather save for one of your big items. Maybe if you plan on doing a lot of early pushing. Up to you. But you definitely don't want to build Hand of Midas. Nature's Prophet can farm so quickly with Teleportation between farming pushed lanes and popping into the jungle that it's not really necessary in most games. Once Teleportation  hits level 4 you should be teleporting around the map every 20 seconds, farming, ganking, fighting, pushing. The gold spent on a Hand of Midas could be a Blink Dagger, half of an Orchid Malevolence or Desolator, even a Medallion of Courage and finished boots. All of these are more useful early in the game.

After your boots it depends on the game. Here are the items I like and the situations to build them in:

• Maelstrom: If you're gonna be farming creep waves or pushing, pick this up. The extra waveclear, damage and attack speed are all great. Good as a first item if you aren't going to be ganking right away. Otherwise get Orchid first.
• Blink Dagger: You need more mobility (which is almost always). Should be 1st or 2nd item most games. You can use it to quickly escape danger (as long as you see it coming). When ganking, you can also teleport into trees near the enemy and blink out to catch them off guard.
• Orchid Malevolence: The typical ganking item. When ganking: Blink if you have it, Silence, Sprout, and right click down. This will let you kill most heroes, if they don't have Quelling Blade or Tangoes. If they do then chasing is still easy with phase.
• Scythe of Vyse: If the enemy has evasion, or is still dangerous when silenced, or has items like Black King Bar/Dagon/Force Staff that could present a problem when ganking, get this instead of Orchid. Not a bad thing to have both though. Hex the carry, silence the support.
• Desolator: Usually the damage item you want. The armor reduction to buildings is too good to pass on for pushing.

Ability build

This can vary, but in lane, I like to take Teleportation first, then  Nature's Call, then Sprout. You can take Sprout sooner if you think there will be a kill opportunity early, but don't put more than one point in it because it scales poorly and is somewhat unreliable. From there, get Nature's Call to level 3, then your ult. 1-1-3-1 by 6 always. Then max Teleportation. The lower cooldown means you can always join fights. Finish maxing Nature's Call after Teleportation, then max Sprout - with Wrath of Nature at 11 and 16 as well.

As far as how to play: simple

You need to be watching the map. Constantly. Now that may be simple but it's not easy. You need to be best friends with the minimap. You need to watch your team to see when they're fighting. If they are, you need to quickly assess if you can help, and more importantly WHERE to Teleportation in to best contribute to the fight. Use your ult during all fights, as well. Don't save it for stealing kills - do as much damage with it as you can. You can't underestimate even a level 1 wrath of nature with ten bounces coming into a teamfight. But when your team isn't fighting, you should be farming or pushing. Farm your jungle if all the lanes are occupied/dangerous, but if a lane is pushed out and there's no immediate danger you can safely farm it with Teleportation. Push any lane that's close to a tower, and bring in your treants for the tower push.

Finally, miscellaneous tips:

• When escaping, Sprout yourself so the enemy can't see you and double-tap the Teleportation ability hotkey to teleport to fountain. Not foolproof because of AoE stuns, blink, enemies inside Sprout, etc. But it's usually enough to get you out.
• When pushing across the river, send a treant out in any direction enemies could come from to scout for you. The safety is worty having -1 tree on the push.
• Blink away into the trees when you sense danger coming, not when you're under attack.
• ALWAYS ALWAYS ALWAYS carry a Town Portal Scroll. By using Teleportation and a scroll, you can be in and out of any point on the map in six seconds. Incredibly useful for courier snipes or just escaping when Teleportation is on cooldown.