Create a reference for your performance. This may be as basic as a click track, so you know where to start singing for each track. Use a percussion sound in your DAW set to quarter notes at the tempo you require. Most DAWs have a metronome function that will accomplish what you need. You may wish to add simple piano or guitar chords so you have pitch reference as well. Record this to a separate track.
Set up the microphone and record your lead vocal. Send audio from the click track and the piano or guitar reference track to your headphones. Some singers prefer to hear themselves in a headphone mix, others prefer to have one ear uncovered. If you use the one-ear method, make sure the mike doesn't pick up metronome sound from the unused ear cup.
Prepare another track for your second vocal. Feed the first track to the headphones with the reference tracks. If you know the second part you want to sing, record it. If not, run test passes over the first vocal until you find the harmony you want to add. Continue this process until you have recorded all the voices and beatboxing you want.
Mix your vocal tracks in your DAW. You will likely need to use equalization to make your vocals fit together. Not all tracks will be equalized the same, because you'll want to balance sound across the spectrum. Try taking high frequencies out of low-pitched tracks, mids out of those in the middle and low frequencies out of the highest vocal parts. Mixing requires much experimenting.