While there isn't a visible 'weapon skill' stat, Tommy's proficiency with firearms improves with use. The more you use a particular weapon type (e.g., pistols, submachine guns), the more accurate he becomes, and his reload speed may subtly increase. This encourages you to experiment with different weapons and find what suits your playstyle.
The most noticeable improvement comes from simply practicing your aim. The auto-aim system in Vice City is helpful, but mastering manual aiming for headshots will make you far more efficient in combat. Regularly visit the shooting range at Ammu-Nation to hone your skills and try out new weapons. Collecting Hidden Packages also unlocks powerful weapons at your safe houses, giving you access to better gear.