How do I remove the blade carriage from the bottom of the mower?

You did not give us much info so my answer will be very generic. Lower the unit as far as it can go. Remove the belt by first disenagaging the belt and retracting the spring loaded tensioner pulley. Peel the belt off the largest pulley by prying and then rotating the shaft. The deck hangs on at least 4 links held by pins and clips. Remove the pins with your finger and push the pin out through the hanger hole. Once all 4 hangers are disconnected the deck is free but you may need to raise the front of the tractor to pull the deck out.

I don't know if you are referring to the deck or the spindle assembly. Since ballengerb1 done the deck, I will do the spindle assembly.
Remove belt from pto pulley at engine, remove belt guards on deck. Use a block of wood to "wedge" the blade against the deck, and remove the nut at the top of the pulley. Remove pulley, remove the bolts that are around the perimeter of the spindle housing, the spindle assembly then will be free of the deck. Some have a ring around the bottom side also. The ring will also be loose. Pull blade and assembly out from under the deck. Remove blade, and replace the entire spindle assembly. Again, ballengerb1 and myself are just kind of out here, not knowing exactly what you were asking.

How do I remove the blade carriage from the bottom of the mower?
Remove 5 finger pull pins; 4 at each corner where deck connects to frame, and 1 in the center linkage(easier to get to on left side of deck).