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Having a distant family branch in which you keep inbreeding (not inbreeding but breeding in) bloodlines.

Explanation:
1. Have a daugher, matrilinearly marry her to one of the Karlings.
2. Have one of their daughters marry matrilinear one of the Irish bloodlines
3. Have one of their daughters marry matrilinear the Welsh bloodline
4. Have one of their daughters marry ...

When you got all the bloodlines you want, have your son and heir marry matrilinear one of the daughters with all the bloodlines. Children will be of her dynasty, but as you started all those generations ago with one of your kids, her dynasty is your dynasty
After 5 generations you should be save from inbreeding
 
I am still pretty new to getting bloodlines. What is the best way to get multiple bloodlines for your characters?

I'd recommend the bloodline breeding guide in my signature. It's extremely detailed and comprehensive. :) If you can't see my signature you can find the guide here. :)

But in short:

Most bloodlines can normally only be passed down by a male, but in matrilineal marriages most bloodlines can be passed down by either parent.

Because of this you can marry a female of your dynasty matrilineally to a bloodline bearer, and the child will have the bloodline and be of your dynasty.

Women of your dynasty that are not too closely related can be married matrilineally to your heirs or yourself to bear children with their bloodlines. :)
 
I am still pretty new to getting bloodlines. What is the best way to get multiple bloodlines for your characters?

Byzantium and heir designation religions are pretty strong bloodline wise as you can bloodline hop and forge multiple bloodlines before breeding into one bloodline