Powering AtTiny85 with Adafruit NeoPixels
The Adafruit NeoPixels already provides a solution for your problem.  

Don't try to run all the power through your AtTiny85.   



Send the 5 volts to your AtTiny85, but branch off the 5 volts before the AtTiny85 and send the power to the VCC (pin3) and VDD (pin5).  

EDIT :  If you are placing the NeoPixels a long distance from your AtTiny, you could run a separate wire (VCC for controls) alongside the power wire (VDD for LED power).  This way you will have a full 5 volts for the controls circuit, not messed up or glitched by power fluctuations by the higher power LED circuit.