Is there a way to know how many enchantment charges are left in a weapon? If so, how?

There's a grey bar underneath the weapon stats in the menu that shows you how many charges, it also appears on the HUD when in combat, above the Stamina bar.

Thank you! So, no number, but a bar indicating the relative number of charges?

I don't believe so.. You get a pretty good sense of how much a strike takes off the meter though.. and a good sense of when it gets "Oh [censored]" Low.

You can see a numerical charge count if you look at the item at an Enchanting table.