Battle points may be obtained by beating games in the Battle Frontier, such as the battle tower where you take on 7 trainers with three pokemon. You could also do the battle castle.
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TM 51 if your using heartgold or soulsilver, you can get it at the battle frontier for 64 battle points
Yes, you will be able to buy a Razor Claw in exchange for Battle Points but in Pokémon HeartGold and SoulSilver, you need 48 points in order to buy it.
AncientPower is not a TM. You can move tutor it to a Pokémon at the Battle Frontier for 40 battle points though.
You get a razor fang at the battle frontier and it costs 48 bp points You get bp points while battling the trainers at viridian city
In Pokémon SoulSilver and HeartGold, the TM81 X-Scissor can be obtained at the Battle Frontier Shop for 64 Battle Points.