In what way would we use "pleading the blood of Jesus"?

Primarily as a prayer for protection. For instance you know that a friend is going to Iraq, You would pray that the "precious blood of Jesus would protect them wherever they go". Or more specifically if you know more specifics.

It is commonly used in exorcism or praying with people who are harassed(but not possessed by evil spirits).

Give Jesus' blood a rest already.

Love and blessings Don

Sometimes we use it just as a reminder to God of His promise that we are His if we are covered by the blood of Jesus, in other words we came to a place in our life where we accepted Christ's death on behalf of us. Not that God has to be reminded, but we claim His promise again at a time when we desperately want Him to act on our behalf, or someone else's. It may seem foolish at face value, but God gives the example of the man who continually bothers his neighbor for bread, and the neighbor finally gives in just because the man is so persistant. Evidently it's o.k. with God to keep bothering Him, and to remind Him we have a claim on Him. And nothing pleases God more than to credit His Son with His work on the cross because of His love for mankind. This was not only a great sacrifice for Jesus, but probably a greater sacrifice for God, Himself. For us to appreciate the value of His Son's blood and all that means has to be pleasing to God. So, the expression is common among Christians in prayer for someone.