Strength is a stat that makes you stronger and is required for some jobs, such as the police. In order to marry Tiffany you must have a high Strength stat.
There are many ways to increase Strength:
Going to the gym (which involves spending money and time)
Fighting people (in which you risk being caught by the police and not gaining any Strength)
Random events that happen while you work (really really random events)
The best way to gain Strength would be fighting people or fighting in any of the bad guy zones. However, there is a limit to the number of people you can fight, and the bad guy zones take 1 week to reopen. There is a person you can have unlimited fights with, though. Once you've unlocked the Chainsaw Club, there will be one NPC that you can fight to gain 10 Strength per fight.