My '77 era two speed Barlows required required a hex key to remove the drum release screw, which you'll find a the bottom of the winch socket. You cannot remove the winch from the boat without lifting off the drum to get at the heads of the mounting bolts.... which are slotted, flat head bolts. That's why you'll want to carefully remove the drum, because the spring loaded pawls (which create the "ratchet" action) can easily become dislodged. The pawls are small wedges that rest in a form fitting cavity on the spindle, which is inserted through the drum's shaft. The pawls are what keeps the drum from rotating backwards. If there's grease or sludge on the pawls, they may stick to the drum when it's lifted up... you only have to see this once to actually understand the issue.
After the drum and pawls... go ahead and remove the cylindrical roller bearings and all the other moving parts... I have never used the box with a hole method, I'm sure it's effective and safe, what I've always done is place a towel around the base to keep any dropped part from bouncing. I will tell you that if you're careful you shouldn't have a problem... but the first time you do this you will not really know what to expect so it can be nerve wracking...
As far as maintenance kits.... definitely invest in one. Lewmar or Harken... the tube of grease will last a life time, cause you don't use that much.. I have six winches on my boat and I still have the tube of grease the previous owner included 20 years ago... The kits have instructions, a small applicator brush and some light weight machine oil for the pawls (never use grease on the pawls) ... replacement pawls and springs may have to be purchased separately. But I have found the Lewmar springs will work on the Barlows fine...the pawls will most likely match up too.
So... follow the instructions that come with the kit... or watch one of the zillion vids. The big difference between the various makes of winches is how the drum is removed.... so... once you've figured that out everything else is universally common. good luck