Grease has always been grease as far as I am concerned, I just use "disk brake" grease that is slightly more expensive and does a better job with heat. Technique for getting it into the birfield is to remove the birf shaft (slide axle into a long pipe and bang down till it pops out, or some other similar mechanism). I then put a deep socket on a long extension, with the socket backwards, then use it to jam new grease into the birf so it is literally oozing past the balls and the birf is completely packed with grease.
You also want to replace the grease in the wheel bearings too. Techique, with your gloves, is get a good maybe 1/2 golf ball goop in your hands with rubber gloves on of course, then you sort of cup the sides of the bearing over the grease so you literally completely jam pack the bearing, same thing, so old grease is oozing out. This is also a great way to remove the old worn nasty grease... To get to the inside wheel bearing, I use a pry bar and literally pop out the old seal then remove the inner bearing. The rebuild kit should include a new inner hub seal (the hub is technially the big part that the disk brake is attached to, that holds the bearings and spins), pack that one, and lightly tap the new seal back in.
Hope it helps!