By playing as the Byzantines, I learned about some of the Byzantine army's weaknesses. Basically, they are very weak against cavalry. Thus, you might want to modify your standard army - lose some of those infantry units (particularly the spearmen) and load up on cavalry.

It sounds like you're playing as a Catholic faction, so be sure to bring some heavy Catholic knights with you. They will be able to chew through Byzantine infantry with ease. However, do not take on Varangian Guard with your knights, except to deliver a final charge against an already-weakened unit. VG will take losses against knights, but their excellent combat stats will eventually give them the victory. The only weaknesses of VG are that they are slow-moving, and usually do not appear in large numbers. Thus, try to isolate the VG, deal with the rest of the army, and then swamp the VG with armor-piercing units, like some good polearms. VG have excellect morale; expect them to fight to the last man and inflict significant losses on your troops.

Against Kataphracts, polearms are effective, as others have pointed out. Also, Catholic knights heavier than feudal knights can defeat them most of the time, especially if you support them with something else. Kataphracts are slow-moving for cavalry and vulnerable on the flanks; try using the superiority of your own cavalry's mobility to outmaneuver them in a cavalry battle. If you are attacked by a cavalry-heavy Byzantine army, try defending in the woods with polearm troops.

So basically, you need to modify your army. Instead of swords/spears/archers/medium cavalry, you should load up on heavy cavalry and polearms. I should note that halberdiers will mop up both Byzantine infantry and Kataphracts, but they will lose against VG. Spearmen will probably be almost completely useless, since they will be easily beaten by Byzantine infantry and VG. Catholic swordsmen will also have a tough time, since they will be significantly outnumbered by Byzantine infantry and outmatched against VG and Kataphracts. Armor-piercing missile troops can be useful, but only if they get a chance to shoot for a while. Your traditional army stands little chance; adapt and live