There are literally countless ways of building up a puppet's body, each with its own advantages and disadvantages. Professionals - who need to duplicate puppets - often use casting methods, whereby they'll sculpt a prototype body, make a mould out of this, and then place armatures in the mould and fill it with foam-latex compounds. This produces great results, but can be somewhat daunting for a beginner. I will therefore go over a simpler process which I have found to work well enough for one-off puppets.
When building a puppet's body, the reference you should try to keep in mind is that of a fairly muscular (human or animal) body: it needs to allow the armature to do its job, and help rather than hinder it. This means that it should be flexible without being snappy (i.e. when an elbow flexes, you don't want the skin to oppose the flex and attempt to set it right again). It should also be relatively lean, because soft areas in your puppet may deform and show up when you animate it.
Foam-latex is perfect for this because it combines the softness of something like Polyurethane foam, with the elasticity of latex. But if you don't have any and would rather not bother making it (because it can be a tad tricky) then you could alternatively use liquid latex along with some easier to procure materials. The process described below is what I have used for the puppets in A Shroud of Silence.
1. Cover up the armature in masking tape
Regardless of what kind of armature you are using - wire or ball-and-socket - you'll generally want to cover it up first. Not only does this prevent wire armature from unravelling, but it also protects all armatures from getting clogged up with latex and whatever other materials you will use in the creation of the body.
Heat-shrink tubing can be very good for covering the armature, but it is not absolutely necessary. You can just use a layer of masking tape, which will later absorb the liquid latex to become flexible. Just make sure you don't use too many layers of tape, as that may potentially impede the flexibility of your puppet's joints. Remember that limbs will need to move, so you need to be careful with anything which might obstruct this.
2. Rigidify relevant parts using plumber's epoxy
As mentioned earlier in the armature section, you'll generally want to rigidify the areas of a wire armature which correspond to human bones, so that when moving a puppet's hand, it bends at the elbow rather than the forearm. The rule of thumb is: if you don't need it to move, make it rigid. And to do this, one simple method is to use plumber's epoxy to cover up parts of the armature and essentially create bones.
Plumber's epoxy is a two part putty which - after mixing together - will solidify within a few minutes. Keep in mind that it really does harden very quickly, so the trick is to work in small pieces, and never mix too much of it at once. Also note that after solidifying it becomes rock-hard, so it will be very difficult to make any more changes to it. So make sure not to use too much, because you really don't need to - even a thin layer is enough to provide adequate rigidity.
One other thing to consider at this point is the puppet's foot. For most tie-down systems you will generally want to use plumber's epoxy to encase them in the foot, and make sure they are firmly secured to the armature. But depending on whether or not you expect you'll need to animate the puppet walking, you might want to make the foot out of one or two separate solid pieces. If the puppet doesn't need to walk, then having the foor be one single solid piece (like in the picture above) is easier and more durable. For walking, though, you'll need to leave a flexible part between the sole and the heel.
3. Cover up in latex
Once you are satisfied with your puppet's skeleton, and that everything (including the tie-down system and the head) has been attached using the plumber's epoxy, it's time to move on to the soft parts. At this stage I like to begin by adding another layer of masking tape over everything, as it will help the latex adhere better.
Liquid latex can be found at craft stores, and is commonly used for casting or prosthetic makeup. It's generally thick enough that you can use a brush to apply it to the puppet. Avoid dunking the whole puppet in the jar of latex, and instead just take your time to brush it on to one side, wait for it to dry, then repeat for the other.
First thing to note here is that you'll want to use a dedicated brush for this - the latex will ruin it. The other thing to note is that once dried, the latex will turn yellowish (as opposed to the white you saw while applying it) and it will be very smooth. Applying subsequent layers of latex on top of each-other will be very tricky (as it tends to just run off), so if you need to do this, then consider applying more intermediate absorbent layers such as masking tape or ladies' stockings.
4. Add muscles and volume
After these first steps your puppet should have some skin and bones, but not much meat on them. Returning to the earlier point about avoiding softness, you really can't rely on costumes to cover up your skinny puppet. Any space between the puppet's body and its clothing may lead to deformations which look bad in the animation stage. You should therefore aim to fill up the entirety of your puppet's intended volume.
Now if you were to use only latex for this, the result would be far too rigid (having a density similar to a car tyre). So you want to use latex only as an additive, to "rubberize" other materials. Polyurethane foam can work well as a base material (which you then drench in liquid latex), but so can textiles. For the puppets in A Shroud of Silence I have used elastic bandages to act as muscles, and felt for areas which didn't need to flex. These can be held in place using masking tape or universal glue before applying the final layer of latex.
