Certain unique characteristics of Christ make Him the great and sufficient High Priest for us today. Note the points made by the Hebrew writer in chapters 4 & 5.
Jesus passed through the heavens (4:14). This makes Him great. His ascension and exaltation (cf. “the firstfruits”, 1 Corinthians 15:23) indicate that His position emanates from the Father in heaven.
Jesus can sympathize with the weakness of men (4:15). All have been tempted and sin (cf. Romans 3:23). Jesus is uniquely qualified to offer a sacrifice for sin because of His Deity, as seen in the preceding point. But also as One who has been tempted like we have, “yet without sin” (5:15).
He was appointed by God. (5:1-4). No high priest can simply appoint Himself to the position. In the Old Testament, God appointed the tribe of Levi and the house of Aaron for this important work. The High Priest was the one to offer the sacrifice for the sin of the people. Under the New Covenant the Father appointed Jesus, after the priesthood of Melchizedek as uniquely the High Priest for all of humanity (5:5-6).
Jesus is uniquely qualified and has been perfected by His death, resurrection and ascension. He has become the “author of eternal salvation to all who obey Him, called by God as High Priest…” (5:9-10).