Brand is a caster mid champ that is normally only found to play as the AP-Carry on mid lane. His passive - or better said the additional effects his spells gain while his passive is on a target - will make u want to target your enemies with your extremely high burst combos instead of more constant harassment. On the other side keep in mind you'll want your spells for farming.
To neither leave out one of these issues you should try to set your Pillar of Flame in a way that both targets your enemy champion and his caster minions. As you won't always be able to set Pillar of Flame in a way it hits both your enemy and the minions you should have a short look on your screen to see how many minions are still there to block your ability to hit with Sear for the stun.

There are different oppinions of how to build Brand's rune pages - I personally request focussing on ability power in order to increase your damage. Yellow runes can be build on armor or maybe scaling armor. Brand does not need any specialiced rune pages that differ from you favourite ap-runes like Akali.

There are basically two ways of efficiently building Brand's mastery pages but they both skill 21 points in offense and then building either defense until Veteran's Scars or Utility until Runic Affinity .

As with his masteries Brand can build his skill sequence in two different ways. I perfer to level Pillar of Flame first and max Sear last - but take an early point in it for the stun. Conflagration is great @ farming and Brand's only aimshot, but it doesn't do the same dmg as Pillar of Flame so I max it second.

You can also combomax some skills as you want or need it.

Brand really needs some quite expensive items like Rabadon's Deathcap so farming should be very important to Brand players. As I already mentioned in the intro you should try to find a balance between farming and harassing the enemy. Especially in your early game you should not really pay attention to use your spells in a way so that they only harass the enemy but concentrate on farming - when you use your spells. On your first 2 levels try to save your mana until you've bought some regeneration items - but don't hold it all back. Instead of the 3 Heal Pots, 2 Heal and one Mana Pot should help you better saving your mana in lane.
Brand's farming is extremely effective once you've got Rabadon's Deathcap but beware...due to your missing escape and dure ability you should not push too far if it's not safe. While you concentrate on farming mostly (but not only) with your auto attacks in early game you'll soon start to use your spells to farm minions (especially the caster minions). When farming with your spells you should try to set your Pillar of Flame so that it hits both the caster minions and the tanky minion to hit the tanky minion with Conflagration afterwards. This allows you to hit the melee minions with Conflagration as well - and sometimes even your enemy champion. After you've bought an Amplifying Tome this two spell combo can be used to kill the caster minions without further spells, hits or the dot from Blaze. Generally you can say: surviving > farming > harassment but this should not hold you back to try to get some kills when they seem safe or when you want to kill your enemies.

Brand's biggest advantag is is incredible burst damage. Any low life mid champ like Ahri, Veigar or Karthus looses more than half of his total health when you once hit a combination of your 3 standard spells. Normally you'll try starting with Conflagration, land Sear second to hit the additional stun that secures Pillar of Flame to safely hit with it's bonus damage. Your ultimate and Ignite should finish your enemy off if you still need it. In team fights try to stay back and help your team with your exceptional aoe damage.
One of your advantages against other champs is your enormous range especially with Sear and Pillar of Flame. When running away you can try to hit your spells from range and probably stun the first persecutor but watch out for long range gap closers - trying to stun is not worth getting caught by Blitzcrank's grab or Lee Sin's Q. On the other side when chasing enemy champions try to set Pillar of Flame in front of them to prepare for Sear to stun but be aware your opponents might know what you're planning especially when you're repeating this combo multiple times (this doesn't need to mean you won't get them anymore ;-))

What you need to pay most attention to when playing Brand is not getting caught by other champs because your only escape ability is your Flash and your only crowd controll (the combination of any spell plus the stun of sear) takes too much time in close running away. When you get focussed in team fights Zhonya's Hourglass is your best friend if you've got it. As with most ap champs you're in danger once an enemy champ focusses on you.

Your item build should focus on inflicting as much damage as possible with your spells. Start with Boots and pots (3 health and 1 mana should do best). At the moment i love to start with Doran's Ring and 2 health pots. When returning to base first time I recommend building mana regeneration. Tear of the Goddess should do great for that. You can also try to build Chalice of Harmony, Doran's Ring, etc early on. As Brand is quite vulnerable I recommend to build some tanky items like Rod of Ages, Rylai's Crystal Scepter, Liandry's Torment or Zhonya's Hourglass. But though building tanky keep in mind that you should still be able to deal some damage to your enemies so tanky normally does not mean to build items like Warmog's Armor, Thornmail or Sunfire Aegis - for Brand.
The pure defensive items you might consider to buy is Guardian Angel or Banshee's Veil but that's when you wanna play really defensive.
If you have problem's in farming (generally or caused by your enemies) consider an early kage's lucky pick (the only gold item that makes sense to Brand) that you'll increase as one of your last item to maximixe it's gold increase.