A quick google search tells me that a Battle Mage is not ideal however...
Dragonborn Sorcerer taking Draconic Bloodline path with melee fighting style.
- d6 dice means low hp. You might need to make up for this with a decent Con score.
- You aren't proficient in any armor. Being a mountain dwarf would fix this, or you could simply get the feats for light armor and so on. You said you're dragonborn though so you're going to need some Dex to work with.
- You want to go Strength type attacker, which is fine except it splits up your stats even more. Also the only weapons you would have proficiency with that'd work for this situation are quarterstaffs and daggers.
- Two types of damage resistance!! Think hard before you choose these, as your racial benefit is a permanent resistance, but your class feature also increases a certain type of damage. Definitely get Fire for one of them, but the other type should be whatever you think you'll have the most trouble with.
- Increased hp with class path.
- Your AC automatically becomes 13 + Dex with Class path. With half decent Dex you'll actually have good AC.
- Having proficiency isn't necessary with weapons, it just means you lose your prof bonus. So you could use a freaking glaive at level 1 if you want. The prof bonus eventually becomes really good though, so definitely grab the Weapon Master Feat at lvl 4.
- Quickened Metamagic!!!!! Attack an enemy, then cast a spell, all in one turn~ This makes up for not having Extra Attack. In general, I'd say you shouldn't just "expend all your magic" then run in. Use your attacks and magic in tandem with each other.
- Constitution and Charisma Saving Throws. This is actually a pretty decent combo. It doesn't have the advantages of Dex or Wis, but those can be made up for with spells, feats, or smart play. The Con save though makes concentration checks WAAAAAAY easier. great for melee spell users.
Note: In general I discourage multiclassing. You'd miss out on great Sorcerer Features, and it makes your character a little messier. Not to mention, dabbling in something like Barbarian only gives you 2 rages per day. Not that great or useful if that's something you rely on.
STR | 16
DEX | 13
CON | 12
INT | 8
WIS | 10
CHA | 16
Skills: I'd suggest Intimidation, Athletics, Arcana, and whatever else.
- Daggers - good for throwing, can use STR for bonuses. Dual Wield with ease. I'd always keep a few on you.
- Glaive/Halberd - Great choice, especially early game. You can be this crazy damage berserker while still adapting more of a hit and run style or by staying behind tankier allies at early levels. No matter who you are, it's real easy to die at level 1. Staying 10 ft. away from enemies helps mitigate damage.
- Greatsword/Maul - Skip the tactics, hit as hard as you can. If they die then they can't hurt you anyway.
- Quarterstaff - If you're itching for the proficiency bonus, go ahead and use this as a versatile weapon for the 1d8.
- All single handed melee weapons aren't effective because your bonus action is reserved for spells and you can't hold a shield. Stick to two handed weapons for the most part.
Weapon Master: +1 Dex, making it 14! Get proficiency with your 2H weapon of choice, javelins for throwing, and maybe something that you can dual weild weild with like Short Swords. Rest is up to you.
The rest of your ability score points should probably be spent on Charisma and Strength. Getting more Con is also a good idea since you'll be making a lot of con checks.
Metamagic: Quickened is a MUST, Extended Spells is also useful because you'll want to use buff type spells a lot.
For cantrips, get a ranged spell, like Fire Bolt or Ray of Frost - preferably one that deals damage of your Class Feature's Dragon blood. Rest is going to be whatever utility cantrips you find useful.
False Life, Expeditious Retreat, Feather Fall, Thunderwave, Ray of Sickness, and Shield. I would highly suggest Shield. Ray of Sickness would be very useful in fighting strong melee combatants. The rest are useful as you'd expect them to be - and False Life will save your life at lvl 1.
For second level spells, continue to focus on buff spells (Enhance Ability), defense spells (Blur), and ones that have melee combat focus (Misty Step). Getting Damage spells is a good idea in general, especially ones of your sorcerer type. Did I mention Enhance Ability?
Btw you should get Enhance Ability.
Also, there's this one spell called Enhance Ability, you might like it. Just think about it.
And so on and so forth.
The build I just suggest is more effective than you'd think. Sorcerer's get a lot of spells, and they have a decent kit for melee combat even with low hp. Don't succumb to the urge to multiclass just because your idea for a sorcerer is unconventional. Fight through!
EDIT: For casting in melee combat, remember not to use spells with ranged attack rolls. Any spells that use saving throws won't be hurt by casting in melee.