I've been playing Brand as my main mage for quite some time now. I feel like I perform well enough with the character to start giving out advice on how to be competitive with him. Hopefully, this guide will teach you the basics as well as some more advanced gameplay.

Pros

-Insane amounts of burst damage.
-A 2 second stun built into the spell combo.
-An ultimate that can just win a team fight.
-Built in ignite on all spells.

Cons

-Pretty squishy before you get a Catalyst.
-Mana inefficient until you get a Catalyst.
-Sometimes you have to wait for some spells to come off CD in order to spell combo correctly.

I believe these to be pretty standard AP Carry runes and masteries.

Laning phase:

Generally, you want to be mid or solo top with most AP carries, and Brand is no different. To start with I generally take Boots of Speed and 3 Health Potions for good mobility for harassing and sustained lane time.

Early game:

I usually rush a Catalyst, but sometimes if I feel like I am dominating the lane and don't need as mush sustainability I will get a Health Crystal followed by a Blasting Wand, and then from there just try to finish my RoA. After you finish your RoA you want to get your boots upgraded. 100% of the time you want Sorcerer's Shoes.

Mid game:

This is when you want to work towards your Needlessly Large Rod. If you can;t seem to keep your farm up you can get a Blasting Wand first, but I would really try to get the Rod. From there you want to complete your Rabadon's Deathcap to complete your core.

Late game:

After Rabadon's it's pretty much fair game to get whatever you want. I prefer going for a Void staff to make sure that my spells keep hitting hard for the rest of the end game. After that you could stack Archangel's Staffs, get a Banshee's Veil, Gaurdian Angel, Morello's Evil Tome, etc.

The reason I max Pillar of Flame first is because it has the longest range of any of your spells, it hits multiple targets (great for farming), and is your highest burst damage spell. I max Sear next as it is the next highest damaging spell as well as the one you will use second most often. I max Conflagrate last because I mostly only use it for farming and to add another spell to team fights after I R->W->Q their team.

Top Tier:

Flash
Ignite

I believe these two Summoner spells to be off the charts in power level. These two spells will allow you to dominate your 1v1 lane and even allow for a very good chance for FB. Using flash to get around minion waves to land your Sear for a stun is often times a kill. Ignite adds to your own personal passive burn to ensure that you get the kills that you setup.

Middle Tier:

Ghost
Cleanse
Clarity

These Summoner spells do not add as much as the top tiered Summoer spells, but they may be suited to your playstyle and are ok options for Brand.

So how do I play Brand?

In the laning phase you want to assert your dominance as Brand is one of the best early solo laners in the game. Conflagrate the enemy champ whenever your creep wave outnumbers theirs and toss in an auto attack. After you hit level two add in a Sear after you Conflagrate them if you have a clear shot and put some really good damage on them (sometimes at level 2 this is even good enough for a kill combined with ignite). After level 3 add in Pillar of Flame if you can get the stun combo off on them for an assured kill. Once Pillar outlevels Conflagrate (level 4) start harassing with it and try to hit the caster minions along with it. Even if your pillar misses you can still get LH on all the caster minions and get some harass on the enemy with a well times Conflagrate. When you are close to level 6, if you feel like your burst can kill your enemy, save enough mana to E->Q->W->R and try to hit R when around an enemy creep wave. With this combo+Ignite there are very few champs who could survive.

The combo mentioned above will be your Bread and Butter skill sequence when fighting in 1v1 situations. The reason for this is Conflagrate doe not do anything to a single target if used after another ability so it is used to start the burning. Sear second as it hits your stun. Pillar third because it causes the most burst if your enemy is already aflame.

For team fights the skill order generally ends up looking like this:

R->W/Q->Q/W->E and then a mix of whatever you need as the fight progresses. The reason you start with the Ultimate is because in a short burst timeframe R+W = highest damage output to the most possible targets. You then want to line up a stun on an enemy that you hit.

Make sure that when getting chased you save spells until you can proc a stun. NEVER NEVER use Sear first in any situation other than just poking/harassing when pushing an enemy tower.

Hopefully, this helped you understand Brand a bit better. This is the way I play him and I usually perform very well with this method in Ranked and Unranked. Brand is very hard to deal with if played well with good farm. You can easily dominate the early lanes with him and run top/bot after level 6 to help get kills with your Ultimate. First blood is your goal, and never forget it.