I've done a lot of work with carbon fiber and glass in canoe building and other projects. It is practical to make a carbon fiber frame sheet starting from scratch. Trouble is, you will spend more than $75. And if this is your first attempt at working with composites, you may waste time and material. I wouldn't recommend it unless you are already familiar with the processes and are ready to spend the bucks to experiment with different layups. I cannot do more than ballpark what would work.
Carbon cloth usually costs about $30 per yard for 36-40 inch widths. But you might consider using a mix of carbon and S-glass - a treated glass that has improved adhesion to epoxy or a Kevlar/carbon mix. More flexible than the glass mix, but even stronger and almost as light as the graphite alone. Both are lots cheaper and perform better than carbon alone. Not as rigid, but less likely to fail catastrophically. Much of the weight is in the resin anyway.
I would use a thin form core - 1/4 to 1/8 inch. It will double the strength. You will still probably need to mold in some structural shapes (like channels) for rigidity.
Here is where I would start: Two layers of 100% carbon cloth ($30) - one on each side of Rohacel foam 1/8" (about $15) Any hobby epoxy such as West System, Eager or System Three (I prefer either of the latter two. You will need one quart of each epoxy component - total 2 quarts. Cost ranges widely - from $35 to $60 for both quarts. I would not consider using polyester resin.
I would mold several test tabs right onto the framesheet and stress them to failure one at a time, adding more material (additional layers) as needed and where needed. When the sheet reached the target strength, I would then cut off any remaining tabs.
Eager Plastics, the Composite Store, John R. Sweet and West Marine are good sources. If you shop around, you can get enough epoxy for about $40. You will need other supplies such as gloves, epoxy measuring pumps ($15), auto repair squeegees.