What are the procedures for appealing a civil lawsuit in state of CA

Unfortunately your question does not include enough facts to provide a good answer. Did you file your complaint after the statute of limitations ran on your claim? If this is the case, there may be circumstances that caused the statute of limitations to be extended, but without further detail, it is difficult to determine your chances of success on appeal. Also, what decision of the court are you appealing?

It is unclear from your question whether you filed your lawsuit or your appeal "one week late." If you filed an untimely appeal, then you cannot win it because in California an appellate court lacks jurisdiction to consider an appeal that was not filed within the time period mandated by statute. If you filed a timely appeal from the dismissal of your lawsuit based on the expiration of the statute of limitations, then you would need to meet with an attorney who can review all of the facts in order to determine whether any tolling period could apply to save your lawsuit and, if so, whether it is too late to raise that issue.