"
Santa baby, forgot to mention one little thing,
A ring
The singer wants to be wed. Whether that is in general, or to Santa, cannot be deduced from the lyrics.
How charlisobel gets to the point that the singer eventually also wants to kill Santa Claus, is a complete mystery. Then again, his (or her) entire murder angle in these lyrics are a complete mystery. Use less imagination, and more heavy medication, I would argue.