Here's an idea: (I don't know your level so forgive me if I over-explain things.)
Put a rectangular mask around half your logo on add. Duplicate and on the second layer change the mask settings to subtract. Key frame the position so that the two are pulled apart
This will give you a clean, horizontal cut. For something more dynamic, you might want to try making the cut-line of the mask diagonal before you duplicate.
*For a more complicated "tearing" sort of look I would to bisect the logo in a program like photoshop so that each half of the logo is flush with the edges of it's comp. Then, I would pull them apart in AE with position KFs while simultaneously animating a warp effect (1 row, 1 column) on each. Warning: the animation for that would be tricky to nail.
Now for the blood. Generating realistic fluids in AE is a fools errand. You can play around with hot glue if you want to go for the intentionally fake/cheesy look. But if you want real blood I suggest you scour the internet for some stock video of dripping paint and the like and composite it in.
Good luck, and hope this helps. If you run into any problems post your project and I'll take a look.