If I was in a car accident 11 months ago and am still sufferering with major back pain and seeking treatments, what is my time limit to file a claim?

The statute of limitations varies from state to state.  Since eleven months have elapsed since the accident, you should have completed your medical treatment and been released by the doctor or declared by the doctor to be permanent and stationary which means having reached a point where no further improvement is anticipated.  If you were NOT at fault in the accident, your personal injury claim would be filed with the at fault driver/registered owner's insurance company.  If you have completed your medical treatment and been released by the doctor or have been declared by the doctor to be permanent and stationary, obtain your medical bills, medical reports and documentation of any wage loss.  Your personal injury claim will include these items.  Compensation for the medical bills is straight reimbursement.  Compensation for wage loss is straight reimbursement.  The  medical reports will document the nature and extent of your injury and will be used to determine compensation for pain and suffering which is an amount in addition to the medical bills.  If you are dissatisfied with settlement offers from the insurance company, reject the settlement offers and file your lawsuit for negligence against the at fault party.  If the case is settled with the insurance company, NO lawsuit is filed.  If the case is NOT settled with the insurance company, you will need to file your lawsuit for negligence against the at-fault driver/registered owner of the vehicle prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.

If you are still getting medical treatment and the statute of limitations is approaching, file your lawsuit immediately because if you miss the statute of limitations you will lose your rights forever in the matter.

As for the percentage charged by an attorney in a personal injury case, this varies  but 1/3  of the amount recovered is what is usually charged.  Some attorneys might charge 40% if the case is not settled and a lawsuit is filed on your behalf.  Again, the amount charged varies among attorneys.