Step 1: Get Your Stuff Together
New tam skin
Adjustable wrenches are useful things to have - it saves having to have lots of different drum keys of different sizes.
Lubricant - I've used WD-40 (in a handy stick form), but KY Jelly and beeswax also works really well. Pick something that won't damage the skin.
New tam skins are available from good music instrument stores. Makes include Contemporanea, RMV and Gope
Step 2: Out With the Old...
Take the lugs off and put them somewhere safe.
Remove the metal tuning ring and put to one side
Remove the broken drum skin.
Whilst its at this stage, why not give everything a good clean and get rid of all that old built up dirt, grease, and muck. Kitchen surface cleaner, alcohol wipes or even good old fashioned soap and water all work well. Don't forget the lugs and nuts - they can build up a lot of grot.
Step 3: Grease Is the Word...
Lubricate the screws on the lugs as well.
Step 4: In With the New...
Replace the ring over the top of the skin. Make sure this is centered as well.
Step 5: Lug It...
At this point, only finger tighten the nuts until you just start to feel tension and the ring isn't loose. DON'T GO TOO TIGHT (not yet).
When all the lugs are attached, start tightening them with your adjustable wrench.
Always tighten opposite nuts together (see tuning plan below). Tighten them evenly so that the skin is stretched to an even tension all the way round.
Don't get nervous about tightening them - the skins go a lot further than you think. Keep going until the tam has a nice sharp loud "crack" when hit.