Adam Clarke on Psalms 51:7

Purge me with hyssop - תחטאני techatteeni, "thou shalt make a sin-offering for me;" probably alluding to the cleansing of the leper: Lev 14:1, etc. The priest took two clean birds, cedar-wood, scarlet, and hyssop; one of the birds was killed; and the living bird, with the scarlet, cedar, and hyssop, dipped in the blood of the bird that had been killed, and then sprinkled over the person who had been infected. But it is worthy of remark that this ceremony was not performed till the plague of the leprosy had been healed in the leper; (Lev 14:3); and the ceremony above mentioned was for the purpose of declaring to the people that the man was healed, that he might be restored to his place in society, having been healed of a disease that the finger of God alone could remove. This David seems to have full in view; hence he requests the Lord to make the sin-offering for him, and to show to the people that he had accepted him, and cleansed him from his sin.