The term "battlemage" can refer to several different builds, but the premise is that you are a warrior who uses weapons and magic, though your armor can be whatever you want.

I'd suggest going for Alteration instead of an armor tree, but picking up the arcane blacksmith perk so you can upgrade any weapon you find. Generally battlemages use One-Handed so they can cast spells in their off hand.

I made an example build for you

Let me start by recommending a Breton. Not only do they start with boosts to some useful skills, but their 25% resistance to magic and their racial power are both awesome. If you add the Lord Stone, Alteration perks, and the reward from the "Agent of Mara" quest your magic resistance can reach the cap of 85%. This allows you to handle spell-casters MUCH more effectively than you would be able to otherwise.

I'd also like to note is that ice spells and runes are great for a battlemage. Ice spells will quickly deplete an enemy's stamina so you can avoid their dangerous power attacks, and if you use the Ice Cloak spell you can deal constant damage at melee range while focusing on your sword attacks. Runes are amazing, as they allow you to do good damage to a group of enemies, and with Restoration and Alteration you even have access to paralyze and poison runes. They are essentially magical sniping tools that remain effective in short-range combat. Don't worry about undead resisting frost though, as with Restoration you will be able to cast offensive Sun spells if you do the Dawnguard DLC.

Speaking of the Dawnguard DLC, I'd recommend going to the Soul Cairn and getting the Conjure Mistman spell. Mistmen use ice spells and can also engage in melee with some minor effectiveness. The Conjure Wrathman spell is also amazing, as they are essentially tanks useful for distracting enemies.