Out of all the more advanced characters with flashy combos, Balrog is probably the easiest to learn.
The key is learning how to combo from his cr.LP. His crouching jabs link into almost all his other moves (check his Trial Challenges for examples), which lead into Buffalo Head, which lead into his Ultra.
That's the key. Once you make his crouching jab a threat, opponents will cower up and start blocking low, at which point you can mix in throws, overheads and Focus Attacks. Learn how to play offensively while keeping charge for special moves and Balrog is a monster.
Dash Straight (back, towards + P)
His regular rush punch. It's only safe when ranged properly, otherwise it can be hit by reversal Supers or Zangief's Spinning Piledriver.
Dash Low Straight (back, down/towards + P)
Like his Dash Straight, but it hits low. Mix up between this and the overhead version (hold P), so the opponent won't know whether to block high or low. In addition to that, the overhead version can be linked into crouching jab, which can then be cancelled into Buffalo Head and followed up with his Ultra.
Buffalo Head (down, up + P)
Balrog's headbutt not only gives him a good move for getting off the floor with but it can also be followed up with his Ultra. Just charge the headbutt so the motion is down/back, down/up + P and you'll still have the charge back motion needed for the Ultra.
Crazy Buffalo (back, towards, back, towards + P)
You can cancel into this off any Dash punch, which means it is a good way of ending combos when you haven't got Ultra for the usual Buffalo Head > Violent Buffalo ender.
Violent Buffalo (back, towards, back, towards + PPP)
This is Balrog's biggest threat, as a single connected crouching light punch should eventually lead into this Ultra for monster damage. The easiest way to connect it is to land a Buffalo Head first. Remember to do the Buffalo Head by charging down/back (not down) and pressing up/back + P (not up + P) so you
keep the back charge needed for the Ultra.
Basic Combo: cr.MP xx Dash Straight xx Crazy Buffalo
Advanced Combo: cr.LP > cr.LP > cr.MP xx HP Buffalo Head > Violent Buffalo