Amplifier overdrive and distortion is usually achieved by signal clipping--pushing the gain until the signal peaks are chopped off. "Real" tube overdrive isn't possible without major changes (adding additional preamp stages, etc.), but here's another option:
Preamp clipping can be simulated by installing a diode-clipping circuit. In fact, some big-name amp manufacturers (Marshall and Fender, for instance) have used diode clipping for a more ballsy sound (especially at low volumes.) In fact, it's made a bit of a resurgence in boutique amps...
The danger of converting a tube sound into a harsh "solid state" sound exists. Tubes clip with a softer "shoulder" and have a more pleasant, warm sound....of course, it's a matter of taste, after all.
But a carefully designed diode-clipper circuit can mimic those tube characteristics. "Peak clipping" more closely approximates tubes than "crossover clipping," which is common to transistor design. Plus we can utilize asymmetrical clipping to emphasize the "correct" harmonics. And since our new sound is contained within a tube amp, it will help to soften the effect.
Of course, diode-clipping can be used (and is) for solid state amps, too.
Video clip shows the three switch-selected settings: 1) clean; 2) medium clipping; 3) max clipping:
Warning: This is a "mod," and does require a bit of rewiring. All the usually cautions about discharging power supply filter capacitors apply here. To restate them:
Don't touch the internal components of a guitar amp without first draining the filter capacitors. Please. You could really harm yourself. Here's a link to my older amp project, and the section Discharge those Caps!