You can use a tool like PowerStrip , Custom Resolution Utility or AsTray Plus to squeeze more pixels into your native display resolution. It will be blurry though, and you may get some flickering.
The better method is gonna cost you - replace the display itself for one with a higher resolution. You may need to also replace the display cable and (not likely) the CCFL inverter.
If you decide on upgrading, first find the model of your current display - you can use softMCCS , Aida64, or just disassemble the lappy and look at the sticker on the back of the display.
Then look up the display specs at panelook or a similar site. To change just the display you'll need one with the same Signal Type, Voltage Supply and Backlight specs. You may have to change the cable/inverter if you get a LED display, or one with a different (but similar) LVDS signal. Make sure the voltage stays the same though.
Good luck!
P.S. ifixit has a display replacement guide.