Hello! This is a build for Brand players that don't mind being somewhat squishy and just want to dish out tons of damage without running out of mana.

As with any other AP-heavy caster, you should get the AP items right off the bat

Sorcery -> Blast -> Arcane Knowledge -> Havoc -> Mental Force -> Spellsword -> Archmage -> Executioner
What you do with the rest of the points is really up to you, but I suggest these items (in order from highly suggested to least suggested)

Butcher - Extra damage for farming
Destruction - Helps give you that extra damage when attacking a turret
Tough Skin - Safer while farming and jungling
Bladed Armor - Helps push a lane and jungle
Perseverance - Gives a little more sustainability, and you need points in defense for Bladed Armor

start with: Boots, Amplifying Tome, and Health Potion is a recommended optional extra.
Build Tear of the Goddess as soon as possible to start stacking mana to get Seraph's Embrace.
Next build Kage's Lucky Pick to start getting extra gold, then Sorcerer's Shoes.
After you buy Sorcerer's Shoes, buy Blackfire Torch, then Archangel's Staff. You can buy Enchantment: Furor on the Sorcerer's Shoes whenever you want after this, for extra speed when running from champions you've stunned or damaged, to keep distance between you and the enemy.
after that, your build is basically open to any situational needs.
Need Armor? Get Wooglet's Witchcap. Need more? buy Banner the Command
Need Magic Resist? Get Abyssal Mask.
Another decent item to get is Rylai's Crystal Scepter. It gives you a good amount of hp and the slow is really useful for team battles AND solo battles.
If you're bored and really fed, you can piss off the enemy team by buying Will of the Ancients for some spellvamp
Boots -> Amplifying Tome -> Health Potion -> Tear of the Goddess -> Kage's Lucky Pick -> Sorcerer's Shoes -> Blackfire Torch -> Archangel's Staff -> Wooglet's Witchcap -> Abyssal Mask

Get Sear first. It does the most damage and has a long range and is easier to hit than Pillar of Flame. After Sear, grab Pillar of Flame to help you farm minions. Of course your third skill should be Conflagration, because Brand is all about bursting all of his skills in order. I suggested a certain sequence of the skills based on my personal preference, but after you get all 3 its all up to you (you should probably level up Conflagration last, but keep the skills equal)

As you probably know at this point, all skills have a special effect if it hits them while affected by Blaze.
if you're being chased, jump into a bush and wait for them to catch up (be wary of skill shots by the enemy champion) and hit them with Conflagration, Sear to stun, and Pillar of Flame. If they are close to death, hit them with autoattacks. If they start to run away, use Ignite on them, or if near a minion wave or another enemy champion, use Pyroclasm. if they still aren't dead, Blaze and Ignite might kill them. if they aren't dead, you can chase them and use Flash and Sear to try and last-shot them.

If you know there is an enemy champion in a bush, you can use Pillar of Flame and Sear on the bush. If you use Pillar of Flame and Sear right after one another you can surprise them and stun them. you can then go into the bush while they are stunned and use Conflagration to deal some more damage, and followup with Ignite or Pyroclasm

Flash can give you extra range on your spells to give you that little extra range you need to kill an enemy, or it can be used to escape a gank, since this build makes Brand squishy

Ignite is just a really nice addition to Brand, because Blaze and Blackfire Torch deal damage over time as well

Brand has to be at least at the same level as his enemies, so you should take top lane because there is usually only 1 enemy champion on top lane, making it easier to stun him if he tries to kill you.

My suggestion is to use abilities as much as possible on minions to stack for Tear of the Goddess/ Archangel's Staff. use Sear on the Siege minion, then Conflagration, then use Pillar of Flame to hit as many minions as possible. Try to stay in the lane as long as possible; Brand's abilities allow him to poke the enemy champion out of farming, so use his range to your advantage. If the enemy champion is going to attack you, wait for him to be clear of enemy minions, as they can block Sear, your stun.