If it is any help, I extracted RNA from bird blood using TRIzol. When i'd done this, I treated the RNA in solution with DNase (TURBO by ambion) before passing it through a spin column based kit to clean it, following manufacturer protocols. There might be an algae/plant specific kit available by qiagen, or the pure link kit is very good for most applications
TRI reagent is good and it should work well. Just use a higher amount of algae. Once the algae are in the TRI reagent leave the sample for a few min at room tem. and homogenize well.