No, a mechanics lien would be inappropriate in this scenario. Mechanics liens are meant to protect the right to payment for "construction-related" labor or materials provided.
As far as how to collect the unpaid rent, first and foremost it can be deducted from the deposit. However, if there is no deposit or the deposit isn't sufficient to cover the unpaid rent, filing a suit in small claims court is likely the best option. New Hampshire small claims courts are capped at claims of $10,000 or less. Furthermore, if the claim is for over $5K, you may be required to submit to mediation before the case can be heard.
In any sense, once a judgment is rendered, you can file it in the county clerk's office and request a writ of execution. This will result in the sheriff levying the tenant's property; going after their bank account is your best bet. If the tenant ever paid with a check, you already have the account info.
Good luck!