You can't get unemployment benefits without breaking the law. That's the bottom line. The consequences for defrauding the government this way are pretty serious.
Also, you can't use that money to pay your rent directly. You need to pay yourself through your company. If you don't, you're "piercing the veil" of the corporation, meaning that you are blurring the lines between your LLC and yourself. That means you lose your limited liability, and anyone who sues you can come after you directly in court (instead of just your company).
Unfortunately, starting a company requires capital. Most companies become cash-flow positive (if they ever do) 18+ months after starting. If you can't live off of what you've saved or what you raise for that amount of time, starting a company isn't for you.
That's the unfortunate thing about startups. Not everyone can pursue them. It's not a god-given right. You need to be lucky enough to have a substantial safety net or, otherwise, the ability to save lots of money beforehand.