If memory serves, when doing the TLI burn NASA boosted the Apollo spacecraft to a velocity high enough to essentially escape the Earth's gravitational influence or very near to that velocity. However, it might not be necessary to consume quite that much propellant to reach the moon if we boosted the spacecraft into an orbit like the one used by that satellite where the apogee is out to the moon and the perigee is down to a few hundred miles high. I don't know if that would result in a significant propellant savings without running some numbers but it's possible.

From memory, the Apollo spacecraft were going somewhere around 6,000 MPH when they arrived at the moon. They had to fire the SM's engine to slow down to lunar orbital velocity, somewhere around 3,700 MPH. It'd be interesting to see what the velocity would be as the spacecraft reached the moon. It's possible though perhaps not likely that you'd be going slower than the 6,000 MPH value from Apollo. If true, then you wouldn't need as much propellant to enter orbit or even to land directly on the surface.

Calculating the velocity required to achieve such an orbit isn't really difficult. However, determining the velocity at apogee when under the moon's gravitational influence is more complicated. I might be able to determine that using the equations in "Fundamentals of Astronautics" but my math skills are getting rusty.

Doing a little checking, I found these values based on on Apollo 17

Under "Typical Apollo Burns"
Boost to Orbit: 31,000 fps (21,136 MPH)
Speed in orbit: 25,570 fps (17,434 MPH)
Translunar injection:10,000-10,600 fps (6816-7227 MPH)
Midcourse corrections:150 fps (102 MPH)
Lunar Orbit Insertion:3,000-3,500 fps (2045-2386 MPH)
Speed in Lunar Orbit: 5348 fps (3646 MPH)
Descent orbit insertion: 140 fps (95 MPH)
Powered descent/Land: 6900 fps (4705 MPH)
Lunar launch to Orbit: 6100 fps (4159 MPH)
TransEarth Injection: 3500 fps (2386 MPH)
Midcourse Corrections: 150 fps (102 MPH)
Entry velocity: 36,000-36,300 fps (24,545 - 24,750 MPH)

While doing some digging, I found this page which shows how to model the Saturn V flight to LEO using QBasic. I have not had time to review the code but his working assumptions seem pretty reasonable.