Can the weapons created by blood crystal limbs be used as weapon foci?
If yes, if you use the Tattoo Magic technique from Street Grimoire to create a weapon focus, does the focus still work if the limb isn't actively being used as a weapon?
The weapons created through the Blood Crystal Limbs aren't permanent, they are temporary constructs that the user of the blood crystal has learned to manipulate and use as a weapon. Additionally, the nature of Blood Crystals means that they already have a magical nature, making them less than suitable for use as a telesma. It's never explicitly stated, but I know I wouldn't allow Blood Crystals to be enchanted as foci.
How would you suggest tattooing crystal? That method for adding enchantments to them seems impossible in a pretty self-explanatory method.
Again, personally, I wouldn't allow Tattoo Magic to be used to create Weapon Foci. There would need to be a more definite "weapon" that is being enchanted. The concept behind Tattoo Magic is that the ink and artwork of the tattoo serves as the physical anchor for the focus/quickened spell/etc. Weapon Foci, since they require an actual weapon as their base, wouldn't qualify.
However, this thread isn't here for arguing the logistics of Tattoo Magic. But the long and short of your original question Chummer 5 is Alive, no you probably can't enchant your Crystalline Blade as a Weapon Focus, and almost definitely not through the use of Tattoo Magic.