Don't open a margin account at a brokerage and you will not be subject to a credit check. The only information that a brokerage will look for when you open a regular account with them is whether you have a valid bank account and Social Security number. If you do apply for a margin account, the brokerage will run a credit check on you, but in most cases you will be approved even if you have a terrible credit rating.
Ensure that you have a valid bank account before you attempt to open a brokerage account. As long as you provide accurate banking and personal identification information to your brokerage, you aren't likely to have any issues opening an account.
Deposit money into your brokerage account from your bank account. Pay all relevant fees when doing so. The brokerage firm will check your status with ChexSystems, but it won't run a credit check on your file.
Invest in stocks using the funds in your brokerage account. Your brokerage will not deny you the use of your stocks due to a bad credit rating. If a judgment is obtained against you due to debts, however, your creditors may be able to seize or garnish portions of your brokerage account.
Consider attempting to open a margin account even if you have a bad credit rating. If you want to trade on margin, the brokerage firm generally cares only whether you have available funds to pay it back if you receive a margin call. The brokerage will check your credit rating, but it's mostly just a formality. Students are not allowed to open margin accounts.