Brother,
When in doubt it can be profitable to look at good commentaries:
This is from Matthew Henry's commentary:
Observe, I. How awfully this charge is introduced (2 Timothy 4:1): I charge thee before God, and the Lord Jesus Christ, who [b]shall judge the quick and the dead[/b] at his appearing and his kingdom. Observe, The best of men have need to be awed into the discharge of their duty. The work of a minister is not an indifferent thing, but absolutely necessary. Woe be to him if he preach not the gospel, 1 Corinthians 9:16. To induce him to faithfulness, he must consider, 1. That the eye of God and Jesus Christ was upon him: I charge thee before God and the Lord Jesus Christ; that is, "as thou tenderest the favour of God and Jesus Christ; as thou wilt approve thyself to God and Jesus Christ, by the obligations both of natural and revealed religion; as thou wilt make due returns to the God who made thee and the Lord Jesus Christ who redeemed thee." 2. He charges him as he will answer it at the great day, reminding him of the judgment to come, which is committed to the Lord Jesus. He shall judge the quick and the dead at his appearing and his kingdom, that is, when he appears in his kingdom. It concerns all, both ministers and people, seriously to consider the account that they must shortly give to Jesus Christ of all the trusts reposed in them. Christ shall judge the quick and the dead, that is, those that at the last day shall be found alive, and those who shall be raised to life out of the grave. Note, (1.) The Lord Jesus Christ shall judge the quick and the dead. God hath committed all judgment unto the Son, and hath appointed him the Judge of quick and dead, Acts 10:42. (2.) He will appear; he will come the second time, and it will be a glorious appearance, as the word epiphaneia signifies. (3.) Then his kingdom shall appear in its glory: At his appearing and kingdom; for he will then appear in his kingdom, sitting on a throne, to judge the world.
This is from the "Jamieson, Fausset, Brown" commentary:
[b] in season, out of season[/b] --that is, at all seasons; whether they regard your speaking as seasonable or unseasonable. "Just as the fountains, though none may draw from them, still flow on; and the rivers, though none drink of them, still run; so must we do all on our part in speaking, though none give heed to us" [CHRYSOSTOM, Homily, 30, vol. 5., p. 221]. I think with CHRYSOSTOM, there is included also the idea of times whether seasonable or unseasonable to Timothy himself; not merely when convenient, but when inconvenient to thee, night as well as day (Ac 20:31), in danger as well as in safety, in prison and when doomed to death as well as when at large, not only in church, but everywhere and on all occasions, whenever and wherever the Lord's work requires it.