From Halopedia, the Halo wiki
He was onboard the freighter Finnegan's Wake when the Insurrectionists set off explosives that killed First Lieutenant Canfield and wounded and killed some other ODSTs, but Chesnick (and most of the surviving ODSTs) was saved by the quick thinking of Lieutenant Jacob Keyes.
Afterward, while Keyes was preparing to fly to Scyllion Warehouse District on Charybdis IX, he was tackled by several ODSTs, including Chesnick, Markov, and Faison.  At first they played a practical joke with Keyes, and made him think that they were going to kill him for pulling rank during their mission to the Finnegan's Wake, but they were really holding him down so that Chesnick could give him the Helljumper tattoo as a sign of their respect and gratitude for Keyes's quick thinking on the Finnegan's Wake, which saved them all.
Chesnick presumably survived the subsequent missions that the Midsummer Night went on, as he is not mentioned after tattooing Keyes.