No, if you're not the owner of the mark, you cannot renew an expired trademark. Only the original owner can do so, and even they can only do so within the grace period, which is typically six months after expiry, depending on the jurisdiction.
Once the grace period has closed and the trademark is fully dead, a third party can file a new application for the same mark. That application goes through the standard registration process and is assessed on its own merits. However, the fact that the mark was previously registered does not confer automatic rights on the new applicant.
Before filing for an expired trademark, it is worth verifying whether the mark is still in active use by the previous owner, as common law usage may complicate the new application even after the registration has lapsed.