Arise, O Lord, in thine anger. To call on God to "arise" is to ask him to take action, to lay aside the neutral attitude in which he most commonly shows himself to man, and to interfere openly in the concerns of earth.
To call on him to "arise in his anger" is to entreat him to vindicate our cause against those opposed in us, and to visit them with some open manifestation of his displeasure (comp. Psalms 3:7; Psalms 9:19; Psalms 10:12; Psalms 17:13; Psalms 44:26; Psalms 68:1).
Lift up thyself. This is even a stronger expression than "arise" (Isaiah 33:10). It is a call on God to appear in his full strength. Because of the rage of mine enemies; or, against the rage of mine enemies (Kay, Revised Version).
Force must be met by force. David justifies his appeal for aid by alleging the violence and fury of those whose attacks he has to meet. And awake for me to the judgment that thou hast commanded. The two clauses are not connected in the original, which runs, "Awake for me: thou hast commanded judgment."
The meaning seems to be, "Arouse thyself on my behalf—judgment is a thing which thou hast ordained—surely now is the time for it."