Hitmonchan would be pretty interesting if it had a cool Special like 318, or some neat Speed, but with only Normal- and Fighting-type moves in its arsenal, and no Hyper Beam or STAB paralyze-inducing Body Slams with which to do significant damage to the Psychics that OHKO or 2HKO it, Hitmonchan is just very difficult to use effectively.
Hitmonchan's main use is as a physical attacker, with its pretty decent 308 Attack and ability to run STAB Fighting moves, but it doesn't learn Hyper Beam or any of the damaging kicks that Hitmonlee does.
Send it out against a Normal-type or a non-Psychic or -Ghost with weak Defense and see if you can do some damage.
Seismic Toss is useful in situations where Hitmonchan's other moves just can't do a lot of damage. Counter can work well with the strategy in which you switch into a Body Slam from a Snorlax or paralyzed Tauros and force your opponent to predict a Counter on the switch or high-damage Submission attack.
As far as alternate strategies... it does learn Body Slam (despite its thin frame), so paralysis is nice to hope for, and Double-Edge is also a possibility, considering you could sometimes use that extra 15 power and Hitmonchan has low defenses anyway.
Agility could be useful, but then again, it is too frail to bother with. Double-Edge for power, but Hitmonchan hardly has the HP to handle the recoil. Mega Kick can do a fair amount of damage to Psychics although it has low accuracy. Mimic can improve its attacking options, although Hitmonchan still doesn't have the stats to use even some of the better moves it could get like Recover and Earthquake very well. ThunderPunch (for Gyarados), Ice Punch (for Dragonite), and Fire Punch (for Parasect) are available, but Hitmonchan's Special is too low to take real advantage of them.
Checks and Counters
Hitmonchan can't do much to Gengar and its immunity to Normal and Fighting, but keep in mind that Seismic Toss can hit Ghost-types in RBY. Exeggutor takes little damage from any of Hitmonchan's attacks, as does Slowbro. Zapdos, Alakazam, and Starmie are all stellar counters, but must beware of Body Slam, as none of the above would want to get paralyzed. In UU, Butterfree has a quadruple resist to Fighting and can return a hit with a super effective Psychic, or sleep or paralyze Hitmonchan. Tangela can paralyze it and otherwise tank hits well with its good defensive capabilities.