Like a semi tanky, ap Melee caster mage (not galio). I may have seen an idea for this before but something along the lines of a monk with a Bo staff.
Edit: Holy shit, i just saw this had near 1000 views and was top page of C&C thanks guys.
Morde, Diana, Galio, Kassadin,. Already got melee ap aa based Also Eve
you said not galio after describing galio.
maybe you should just play galio.
To insert a reference I just spent an amusing hour searching for:
"Battle mage? That's not a profession. It barely qualifies as a hobby. 'Battlemage' is about as impressive a title as Lord of the Dance." ... "I am adding Lord of the Dance to my titles."
-Richard: Lord of the Dance, Mistress of Magma, Archmage of Destiny, Lord Ashendale, Mayor of a little village up the coast
Edit: And here I thought I'd be repeating what someone else said but I'm not. How the hell am I the first person to bring up Rumble?
Kind Sir you should specify that you are looking for someone whose main theme is "The Battle Mage" and not someone who just "plays" like one. It's also worth noting that "battle mage" is very loose theme, it can mean many different thing - "mage who specialise in battle magic", "mage who loves to battle", "mage who can win entire battles by himself", "mage trained for battle"... the list goes on. Probably proper theme you are looking for is "The Spellsword" - a character that combines both arcane/elemental magic, some kind of (enchanted) armor and a weapon.
While characters stated by others fit into "mage/warrior" achetype none of them is "spellsword" in its core.
E.g. one could say that 
fits into this theme - "he uses melee weapon and magic hur dur..." - but he is more of "a star/gem battle cleric" in his core (he was "a gem paladin" before his rework but now both Garen and Leona fit paladin theme better than he).
Bo staff.
This peeves me to no end! bō (棒: ぼう), is a very tall and long staff weapon used in Okinawa and feudal Japan. When you say "Bo staff" you are saying "staff staff"...
I looked into Battlemages, and using inspiration from Dragon Age and KoA, I've come up with this skillset. If anyone wants to flesh it out, go for it. :)
Passive: Regains 1-5% (based on level) of his mana when struck by an auto attack.
Q: Empowers his autoattacks to deal damage over time for 5 seconds
W: Blinks to a target location
E: Charges for 3 seconds, then gains a shield for 50-75% (based on level) of the damage taken in that 3 seconds, blasting back 50% of the damage taken as magic damage in a short range aoe
R: Blasts a force of elemental energy in a cone, dealing magic damage and pulling enemies towards him, as well as gaining bonus armor and magic resist depending on how many enemies were pulled
I hope this helps a little bit.
Im thinking a ryze with a more dynamic theme here. You could probably change around his theme/colors and there you go.
A melee mage champion that does more damage with autos based on AP ratios while being tanky would be nice
