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Can I upgrade my computer's CPU from an i3 to an i5 processor?

Welcome to TR!
Most likely, you can. The simplest way is to stay in the series. For example, if you have an i3-6300, you can likely upgrade to an i5-6500 without too much trouble.
To be safe though, you should check the compatibility with your motherboard manufacturer.
For example, the Asus H110M motherboard (https://www.asus.com/us/Motherboards/H1 ... fications/) says

Intel® Socket 1151 for 6th Generation Core™ i7/Core™ i5/Core™ i3/Pentium®/Celeron® Processors
Supports Intel® 14 nm CPU

Then a more specific list: https://www.asus.com/us/Motherboards/H1 ... pDesk_CPU/

Is your computer a laptop or desktop?
What is the current CPU model? (right-click My Computer -> Properties)