Under the New Covenant, what does it mean to "keep my covenant"?
It means to believe that Jesus has Finished the Work:
"Jesus answered and said unto them, This is the work of God, that ye believe on him whom he hath sent." (John 6: 29)
"I have glorified thee on the earth: I have finished the work which thou gavest me to do." (John 17: 4)
Today, we keep His covenant and testimony when we receive the grace of God, giving up on our own efforts and walking in His Spirit:
"Now when the congregation was broken up, many of the Jews and religious proselytes followed Paul and Barnabas: who, speaking to them, persuaded them to continue in the grace of God." (Acts 12: 43)
"20I am crucified with Christ: nevertheless I live; yet not I, but Christ liveth in me: and the life which I now live in the flesh I live by the faith of the Son of God, who loved me, and gave himself for me. 21I do not frustrate the grace of God: for if righteousness come by the law, then Christ is dead in vain." (Galatians 2: 20-21)
"This I say then, Walk in the Spirit, and ye shall not fulfil the lust of the flesh." (Galatians 5: 16)
Paul explains at length that our salvation rests in Jesus, not ourselves or our works:
"8For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: 9Not of works, lest any man should boast." (Ephesians 2: 8-9)
Today, we are no longer under law, but under grace:
"For sin shall not have dominion over you: for ye are not under the law, but under grace." (Romans 6: 14)
The writer of Hebrews explains the New Covenant, and its enforcement clause, and how it exceeds the Old Covenant of Law:
10For this is the covenant that I will make with the house of Israel after those days, saith the Lord; I will put my laws into their mind, and write them in their hearts: and I will be to them a God, and they shall be to me a people:
11And they shall not teach every man his neighbour, and every man his brother, saying, Know the Lord: for all shall know me, from the least to the greatest.
12For I will be merciful to their unrighteousness, and their sins and their iniquities will I remember no more.
13In that he saith, A new covenant, he hath made the first old. Now that which decayeth and waxeth old is ready to vanish away." (Hebrews 8: 10-13)
Today, we live under this covenant of grace, in which He remembers our sins no more, because they were fully remembered in the body of Jesus (Colossians 2: 13-15).
So insistent is this New Covenant on what Jesus has done, that Paul admonished Peter for not walking according to the truth of the Gospel:
"But when I saw that they walked not uprightly according to the truth of the gospel, I said unto Peter before them all, If thou, being a Jew, livest after the manner of Gentiles, and not as do the Jews, why compellest thou the Gentiles to live as do the Jews?" (Galatians 2: 14)
Today, because of the New Covenant, cut for us in the Blood of Jesus (Mark 14: 24), we have His Spirit living in us, teaching us all things (1 John 2:20, 27) and we are seated in heavenly places in Christ Jesus (Ephesians 2: 4-6), and we reign in His life forever (Romans 5: 15-17)
Keep His Covenant of Grace -- believe in Jesus, and grow in grace and knowledge of the Lord (2 Peter 3: 18).