The Byzantines are primarily a defensive civilization that excels at turtling. However they are very versatile and are perfectly capable of mounting a good offense as well.
Strengths
The Byzantines' primary strengths are defense and the ability to effectively counter a variety of troop types (infantry, ranged, and cavalry). Their buildings have bonus hitpoints which increase each age. This makes knocking down their walls difficult and their defensive structures benefit as well. In addition they have access to all defensive buildings and their counter units have bonuses, making attacking them even more difficult. Technologically they are the one of, if not the, most advanced civilizations in the game. They can research almost every technology. In the original game they only lacked the Blast Furnace technology. In the expansion they are missing several others but are still remarkably advanced. They also have a considerable discount when researching the Imperial Age upgrade. A Byzantine military will likely be composed mainly of mounted units as they have full cavalry technologies (excluding Bloodlines), but due to their large amount of technologies can effectively create any kind of army. Their unique unit, the Cataphract, excels at killing infantry but can take on almost any unit due to their large attack and armor. In addition they are virtually immune to bonus damage from Camels and Mamelukes, making these normally potent anti-cavalry units harmless. The Byzantines also possess a strong navy with full naval technologies and have full access to gunpowder units. They also have a faster attacking Fire Ship which helps them defensively in the seas, and makes them slightly better at shore-bombardment in the Castle Age. Also, the Byzantines' halberdiers, skirmishers, and camels have a cost reduction. These units give them effective counters to cavalry and archers, working alongside the Cataphracts' devastating anti-infantry ability.
Weaknesses
Byzantines are versatile, but have a few weaknesses. They excel defensively but they don't have as many offensive bonuses to aid them while attacking, since they lack Blast Furnace. Due to this, Paladins are one of the best defenses against their Cataphracts, since they are also cavalry and possess high defensive and offensive stats. Their siege weapons are also slightly lacking, missing the Siege Onager and Heavy Scorpion.
Although Byzantines have complete Stable unit tree, their Stable units are considered weak because they lack Bloodlines and Blast Furnace upgrade. In one-on-one situation, Byzantines Heavy Camel lose to Paladin with Bloodlines upgrade. The same outcome also apply to Cataphract. This in turn puts them at disadvantage against an army consisted of heavy cavalry. Luckily, this is balanced by the fact that Byzantines possess the cheapest Camels and Halberdiers in the game. So Byzantines are still able to defeat hostile heavy cavalry troops by using numerical superiority.
Strategy
Byzantines have a nearly complete tech tree, allowing them to do pretty much whatever they want, instead of being limited to one or two particular strategies like other civilizations. Unfortunately, they don't truly excel at any strategy save defense. Typically, Byzantines will try to reach the Castle or Imperial Age as quickly as possible, since that is when their tech advantages shine. They are perfectly capable of Dark Age rushing (drushing) or Feudal Age rushing (flushing) if they want to, however. They can flush with any Feudal units, although their Skirmishers and Spearmen are cheaper, allowing them to effectively counter an incoming flush. Maybe the best tactic used by Byzantine players is turtling since all their buildings have a large amount of hitpoints.
In the Castle age, Cataphracts, Camels and Knights work exceptionally well together. All are fast and mobile, Cataphracts help counter Pikemen and Camels, Camels get rid of opposing Knights who can otherwise give Cataphracts difficulty, and Knights are all-around bulky and good for killing Archers and raiding under Town Centers and Castles. Byzantines also get cheaper Camels, making this combination even more effective. With their team bonus increasing Monk heal-speed, the player can also keep some Monks around to make up for the lack of Herbal Medicine.
Castles, Keeps and Bombard Towers make for an effective defense on land maps. On sea maps, Byzantines lack Heated Shot, so they should attempt to win the water with their stronger Fire Ships and complete Dock techs.
One of the Byzantines' greatest strengths in late Imperial Age is their versatile tech tree. Cataphracts can effectively punish any Mesoamerican civilizations or civilizations dependent on infantry (i.e. Goths, Ethiopians, Japanese, and Celts) since they lack proper counters against Cataphracts. Paladins, despite the lack of Bloodlines, can punish civilization with strong archer lines and weaker cavalry such as the Britons and Ethiopians along with their cheap Skirmishers, although civilizations with stronger cavalry Mongols, Huns, Spanish, Franks, Turks, Slavs, Persians, and Magyars are a huge threat to the Byzantines lategame, as their lack of Bloodlines and Blast Furnace makes their cavalry incredibly weak in comparison. However, their cheaper Camels and Halberdiers can help compensate their weaker cavalry lategame.
Strategy changes in The Forgotten
Their new Tech Greek Fire gives Fire Ships +1 range; they already have good and strong Fire Ships so this new tech gives more advantage at the sea and can take down enemy ships for more distance.
Strategy Changes in The African Kingdoms
Byzantines have already a nearly complete tech tree and in this expansion this tech tree is furthered with the addition of the Arson and Arrowslits technologies which in this last case furthers the defensive capabilities of the already strong byzantine towers. Also the Addition of the Fire Galley in the Feudal age gives an early edge to the byzantine navy since they will benefit from the Fire Ship bonus.
Alliances
Byzantines make great support in team games, as they have several defensive bonuses that can aid their allies. Their HP bonus for buildings greatly helps their team, as it allows them to create strong defensive structures. If an ally attempts a Wonder or Relic victory, is recommended that the Byzantines support that player with Walls, Towers, and Castles.
The Byzantine team bonus makes the monk support more effective, as the monks can heal faster. This is specially useful for civilizations that have good monks or bonuses for monks, like the Aztecs, the Saracens (especially when they perform the Smush), the Slavs, the Spanish, the Teutons (which will make their monks the best healers in the game, as their monks have also a healing range bonus that was fixed in the Forgotten).
With an open tech tree and a wide variety of tactical options, most team bonuses will benefit the Byzantines at some point of the game. Maybe the best team mates for the Byzantines are the Mayans, as their team bonus will make the strong Byzantine walls cheaper, the Vikings if playing in water maps, and the Indians, which will add their anti-building bonus to the cheap Byzantine Camels.