All the saints greet you, especially those who are of Caesar’s household.
All the saints salute you, chiefly they that are of Caesar's household. (p) Those who belong to the emperor Nero.
These were slaves and dependents of Nero converted through Paul's ministry in the Praetorian barracks. Philippi itself was a Roman colony, so the Christians there had a natural bond with those of the imperial city, both converted by the same apostle under similar trials.
AI summary
No names are given because the Roman Christians were unknown personally to Philippi, yet they shared a warmth of affection stronger then than now, the first ardor of love, and common suffering binding them closely. The household of Caesar likely means his domestics and court officers rather than his relatives, as conversion among them was more probable.
AI summary