Through the Ages:
The Dark Age:
The Byzantines get a measly +10% hit points to their buildings in Dark, but that is their only bonus. This technically qualifies them as a slow civ, but theres no real difficulty attached to a standard 22-24 villager feudal, or 27+2 fast castle for them - they simply dont receive an economic bonus.
The Feudal Age:
This is where the Byzantines come to life. Youll immediately receive the benefits of Town Watch, giving you much increased visibility around your base. In a feudal war, you want to concentrate your efforts on mass production of your cheap skirmishers and spearmen - the bonus really add up when youre saving 15 resources per unit.
If youre producing from 1 barracks and 2 archery ranges, while maintaining constant villager production, youll save an average of 56 resources per 5 pop, which compares favourably with the Hun house bonus (which saves you around 40 resources per 5 pop). The down side of this is that you receive no bonus on scouts, archers or m@a - while a Hun still would. They also receive no bloodlines for their scouts.
The building hit points bonus also comes in handy here, with house and palisade walls and towers all being 1/5 tougher than their enemy counterparts.
On water the Byzantine are poor in feudal, with their only bonus being slightly tougher docks. Much better to head quickly onto castle for Fireships, or land and put those cheap troops to use.
The Castle Age:
The Byzantine bonuses carry over into Castle age, and if you have significant feudal forces left over, you will strongly want to consider upgrading them. The main addition to your forces should be cavalry. I find a mixture of knights and camels works best - without bloodlines your knights will be inferior to the enemy, but with cheap camels and pikemen mixed in, you shouldnt find it difficult to counter a player making heavy use of their cavalry lines.
Fully upgraded elite skirmishers are a very useful unit to have in castle, effectively countering x-bows, pikemen and horse archers, plus with 5 attack and good pierce armour, being useful as a villager killer.
As you continue to boom in castle age, keep an eye on your resources, as you only require ~650 food and ~550 gold to advance to imperial age. A few Byzantine priests are useful to have around, as they can rapidly heal your injured troops.
The Byzantine navy makes something of a recovery in the Castle Age, with the best fireships in the game, although heavy use of War Galleys is still recommended to compliment them.
The Imperial Age:
Imperial age is where the Byzantines hit their peak, and with a little luck you should hit it several minutes before your opponent. They have access to the fullest tech tree in the game, giving you a lot of options, from Bombard towers to Paladins. The most effective way of using this plethora of options is to counter what your opponent is throwing at you, and continuing to make effective use of your cheap counters units.
Getting bracer and gunpowder for your elite skirms is a priority and this also gives you access to the expensive gunpowder units. Halberdiers is a usual upgrade to get, and you might want to also consider Heavy Camels if your opponent is making heavy use of cavalry.
Your buildings are now also a lot stronger (at +40% hit points), giving you the toughest castles and walls in the game. Think of it like this, enemy trebs will have to fire 40% more shots to take down any of your buildings.
There are two drawbacks however. First of all, the Byzantines can be seen as jack of all trades, master of none as their troop lines receive no special bonuses, and all of them have flaws that effect their performance. They lack Blast Furnace which effects all their melee units, and their cavalry are further hampered by lack of bloodlines. They also lack a few favourite technologies, Heavy Scorpions, Siege Engineers and Treadmill Crane. They do receive every single economic upgrade however.
On water the news is better, you have access to the full naval tech tree, including the now useful shipwright. Although not quite up to duking it out with the Vikings or Saracens, the Byzantines have an extremely effective and versatile late game navy.