Your Microsoft points are tied to your Gamertag, the actual name of your account. Changing your avatar (there is no way to "delete" an avatar, assuming we're using the same nomenclature here) will have no effect on the amount of points on your account.
To check how many points you have on your account, I'd suggest just pulling up the Game Marketplace and selecting a random item. It should display the cost of the item along with how many points you have. (Be careful not to accidentally purchase it though!)
There are likely more elegant ways to check how many points you have, but I haven't used my 360 in months