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In this study, trypsin from the porcine pancreas was immobilized on a heterofunctional support prepared by activating chitosan (Chit) hydrogel with glutaraldehyde (GA), then functionalizing it with glycine (Chit–GA–Gly). The catalytic performance of the immobilized trypsin in the hydrolysis reactions was compared with the catalytic performance of the immobilized enzyme on glutaraldehyde-activated chitosan (Chit–GA) and chitosan hydrogel (Chit). The maximum concentration of immobilized protein on Chit–GA–Gly was approximately 16 mg·g−1 at pH 9.0 (5 mmol·L−1 buffer sodium carbonate) at 25 °C from an offered protein loading of 20 mg·g−1. This biocatalyst exhibited maximum specific activity (SA) of 33.1 ± 0.2 nmol·min−1·mg−1 for benzoyl-DL-arginine-p-nitroanilide (BAPNA) hydrolysis, twice as high as the enzyme immobilized on the classic Chit–GA support (SA values ranging between 6.7 ± 0.1 nmol·min−1·mg−1 and 8.1 ± 0.1 nmol·min−1·mg−1). The Elovich kinetic model was used to describe the adsorption process using low (3 mg·g−1) and high (20 mg·g−1) initial protein loadings. The optimum temperature for BAPNA hydrolysis catalyzed by the immobilized trypsin (60 °C) was 10 °C higher than that of its soluble form. Additionally, the immobilized enzyme was 16 to 20 times more stable than its soluble form at 50–55 °C. Thermodynamic studies were conducted to elucidate the kinetics of the thermal inactivation process of soluble and immobilized forms. Complete hydrolysis of bovine serum albumin (BSA) at 37 °C was achieved after 2 h using a soluble enzyme, while for its immobilized form, the hydrolysis yield was 47%. Reuse tests revealed that this biocatalyst retained 37% of its original activity after 10 successive hydrolysis batches. Based on these results, this support could be used as an interesting alternative for producing heterogeneous biocatalysts with high catalytic activity and thermal stability when producing protein hydrolysates.