Consider whether you have standing to challenge the executor. In most states, only an "interested" person has standing to object to a named executor or challenge the will -- that is, a person who inherits under the will currently in probate or who stands to inherit if the current will is invalidated. A close relationship with the deceased, by itself, does not confer standing.
Investigate whether the executor was actually named in the will. A simple reading of the will is often sufficient. If circumstances warrant, look for evidence that the testator intended another person of the same name to serve or that a third party changed the will terms. Consider whether the will appointing the executor was invalid. The most straightforward challenges to an executor are proof that the testator did not appoint her or proof that the will appointing her was forged, improperly witnessed or otherwise invalid.
Consider grounds for challenging the executor. An executor stands in a position of trust to the testator's heirs; the law imposes the highest possible duty on an executor, termed a fiduciary duty, identical to the duty the law imposes on a parent toward a child and an attorney toward a client. An executor's duty requires that she take whatever action is necessary to ensure that the beneficiaries of the will receive maximum benefits; it strictly prohibits self-dealing or dishonesty. Consider whether evidence establishes that the executor is unfit to undertake that duty or that she has failed to perform that duty.
Be sure your evidence rises to an incapacity if you are claiming unfitness to serve. The court starts from the assumption that an adult is of sound mind, so evidence of unfitness to serve must overcome that presumption. Proof must establish a physical or mental handicap that prevents the executor from performing her duties. Courts disqualify executors -- deem then unfit to serve -- if they have previously served jail time for criminal behavior.
Gather evidence of misconduct affecting the estate, if you base your challenge on improper behavior as executor. Proof that the executor stole or embezzled from the estate, lied to the court or failed to obey a court order generally results in disqualification. However, claims of "estate mismanagement" must comprise more than a difference of opinion about the best investment strategy. Nor do courts often censure an executor for failure to act in the time frame preferred by the heirs. Misconduct must be serious and those challenging the executor bear the burden of proof.
File a timely objection to the will or to the executor with the probate court. Your time frame depends on the probate procedures in your jurisdiction and on the claims you forward. Upon receiving an objection, the court sets a trial date for each side to present evidence. Prepare your case and present your arguments. Convince the court of your allegations. The court either rules immediately after trial or, more likely, takes the matter under advisement and mails a written decision to the parties at a later date.