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.
Strategy changes in "The Forgotten"
Their new Tech Greek Fire gives to 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.