Are you basing them on Codex SM (with Traits?), or on Black Templars (or even Blood Angels)? That's pretty fundamental in terms of organisation and therefore the reasons for the army - or would you rather come up with some great fluff, then pick the rules that best represent it?
What do we know about the Red Templars?
1 - their name (which may suggest a connection with the Black Templars);
2 - their heraldry (which may suggest connections with Blood Angels, but also the Ecclesiarchy in the use of the fleur-de-lys - I think this is right anyway!);
3 - they use bike squads;
and that's about it really.
Heph's right that bikes and crusading are logical - maybe the chapter is a mix of White Scar and Black Templar geneseed, with say 'Be swift as the Wind' (White Scars biking heritage) and maybe 'Uphold the honour of the Emperor' to represent their crusading zeal (Black Templar background); but it's not necessarily the only way to go.
I'd say you have to have a minimum of one bike squad to represent the 'known fluff' - which you can actually get with the 'Eye to eye' drawback (it could be that the picture represents the only bike squad that the chapter posses for all we know!)
They have a chapter symbol which links them to the Ecclesiarchy - why is that? Were they established before the time of Sebastian Thor as a particularly 'religious' chapter, later made independent but retaining the badge that they'd borne before?
Are they 'Red' Templars because their armour is red - or is their armour red because they are 'Red Templars'? In other words, are they 'soldiers of the Red Temple', and if so, what and where is the Red Temple? (I have a loyalist chapter in my fluff called the Jade Templars, whose first company are the 'Knights of the Jade Temple', and whose home-world contains, you guessed it, a temple. Made of jade.)
If they aren't the guardians of the red temple (for instance), does 'Templar' just mean 'crusader'? If so, I'd be tempted to base them on Black Templars (if they can take bikes - don't think I've ever seen BT bikers!)
Only a few notes and thoughts, but I hope of some help!
:cyclops: