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As a successor chapter of the Black Templars I would say that the apple wont fall far from the tree. They'd be very Black Templar-esque in their FoC, but being their own entity you can flavor it to taste.
Aside from that the advice above is solid.

According to the data here they 'are' a successor of the Black Templars founded somewhere before the second war for Armageddon.
http://warhammer40k.wikia.com/wiki/Red_Templars

Ultimately you can do as you like with your project as it were as long as you can field it within a codex.

As to who fluff wise the Red or even White Templars are successors of, both are thought to be from the overly large (5000 ish at last approximation between various crusades), Black Templars. The thought being that it has come to wider attention that Black Templars are not codex compliant in this area and the high lords of terra are siphoning off some of the excess and forming them into separate chapters. But we live in the age of Matthew Ward so who the heck knows for sure about anything anymore.

There is however the case of the Son's of Medusa Chapter being 'recognized officially into being its own chapter after having been a split off group from the Iron Hands that refused to die and instead thrived.

http://wh40k.lexicanum.com/wiki/Sons_of_Medusa

But I think we diverged from the main topic long ago.