Consider whether you qualify to bring an objection to the executor. Most states only allow an heir to challenge an executor, and a familial relationship or friendship with the deceased is not enough. If you are not an heir, work with an heir to disqualify the executor. Explain your evidence and convince the heir to file an objection in her name.
Use precision in identifying any allegations of incompetence. Recognize that your poor opinion of the intelligence or character of the executor will not constitute evidence, nor will vague allusions to recreational drug use or similar habits. Courts presume that the executor is of sound mind, capable of rational thinking, and you must overcome this with compelling evidence of a substantial handicap -- mental, physical or emotional -- that makes the executor incapable of serving. Find evidence that the executor served jail time for criminal conduct or is habitually drunk.
Look for illegal action if you base your challenge on the executor's behavior. The fact that he is impolite, late or disrespectful is irrelevant. The executor owes you and every other heir an extremely high duty of care, termed a fiduciary duty. This duty is equivalent to the duty the law imposes on a lawyer to her client, and obligates the executor to treat you fairly, not courteously. Investigate the executor's actions to find breach of that duty. Look for self-dealing, embezzlement, forgery, lies to the court or flagrant waste. Find evidence that the executor failed to follow a court order or refused to file an accounting. Leave aside arguments that evidence disagreement between yourself and the executor about appropriate investments or timing.
Prepare an objection to the executor and timely file it with the probate court. The time frame depends upon the action you contest and probate rules in your jurisdiction. Alternatively, hire an attorney to represent you. The court schedules a trial date once the clerk receives the objection. Prepare your case, amass evidence and locate witnesses. Appear in court on the trial date and convince the court of your allegations. You may receive a ruling after argument, but more likely you will receive a written decision a few weeks later.