Usually, the Blood Hunter's unarmed strikes are not considered weapons for purposes of their Crimson Rites. A Crimson Rite is applied to a wielded weapon in most circumstances, although there is one Blood Hunter subclass that can add their Rites to their Fists: the Order of the Lycan. Within the Order of the Lycan, there is this ability:
Predatory Strikes. You can apply your Crimson Rite feature to your unarmed strikes as a single weapon. You can use Dexterity instead of Strength for the attack and damage rolIs of your unarmed strikes. When you use the Attack action with an unarmed strike, you can make one unarmed strike as a bonus action.
Your unarmed strikes deal 1d6 slashing damage. The damage increases to 1d8 at 11th level.
So, if you reach Blood Hunter level three and choose this subclass, you will be able to apply your Rites to your Unarmed Strikes. The wording is very similar to the Monk's own Martial Arts feature, which leads me to believe that your DM may find this acceptable, due to the similarity of the abilities, or argue that they cannot stack, also due to the similarity of the abilities.
As always, check with your DM, but going by the wording of abilities, this seems to the be best way to snag the combo without having to argue semantics or look too deeply into the Rules As Written.
I dabbled in writing a few class articles, including one about the Blood Hunter. It's not insanely in-depth, but it might help provide a good comparison and explanation of some of the more confusing or complex abilities you'll receive as your climb the level chart!