Nothing should run hot. It should be barely warmer than room temperature. You should NEVER interface a mirco directly to a device running at higher voltage unless the pins are specifically designed to do so. For example, some PICs work on 3.3V maximum VDD but have 5V tolerant pins.
In reality you have three options:
1. You can power the LCD from 5V but not connect it anywhere to the PIC supply and hope the 3.3V logic levels are enough to drive the LCD inputs.
2. You can drive the PIC pins low, add pull-up resistors at the pins and drive the TRIS bits instead of LAT bits if the pins are 5V tolerant.
3. You can add additional circuitry to convert the 3.3V signals up to 5V.
If you look at the 18F85J11 data sheet pin diagrams you will see some are 5V tolerant and some are not. It doesn't mean the 5V tolerant pins can produce 5V, it means they can withstand 5V if you provide it from outside. So you can use option 2 above with pull-up resistors to the 5V line.
Brian.