Even though Tahm Kench is an unkillable Tank, not everybody may have noticed, that he has pretty incredible AP scalings on his Q(80%) and W(100%). The only problem being, that his W is pretty hard to hit consistently. This means that if you miss your W in a 1v1, you lose more than 50% of your total damage potential plus you don't get the cooldown reduction, leaving you very vulnerable. This problem is easily solved by building Everfrost.

Tahm has nice sustain thanks to his E, so you should be fine taking extended trades as soon as you hit level 3. If you can, try to poke your opponent with Q. Try to only use your W only if you know that you will hit (Amumu gank) or if you know that you are NOT getting ganked, as it is your only way out.
The real fun begins when you get everfrost. At that point, you hit Everfrost, then W then auto attack, then Q, AA, R and walk towards your Turret to spit them out. If that doesn't kill them, it will at least severely cripple them and they will not be able to do much to retaliate.
If you miss Everfrost, you just wait a bit and try again.

The late-game depends on your runes and build. You could go tankier and be a tanky AP teamfight beast, trying to hit multi-man W and eat the enemy ADC, whilst remaining relatively healthy.

Or you can go Straight burst and flank with your high damage output Electrocute and Lichbane procs.
It is important though that you realize you are not your typical Tahm and are pretty squishy.