Demo using Pololu's APA102 library with an ATTiny85 @ 16 MHz.

apa102_raindbow_attiny85.ino

/* This is a fork of the Raindbow sample from 
* https://github.com/pololu/apa102-arduino/blob/master/examples/Rainbow/Rainbow.ino
* to work on an ATTiny85 @ 16MHz. While there is no license in the file, the overall
* library appears to be under MIT: 
* https://github.com/pololu/apa102-arduino/blob/master/LICENSE.txt
*/

#include <avr/power.h>
#include <APA102.h>

// Define which pins to use.
const uint8_t dataPin = 0;
const uint8_t clockPin = 1;

// Create an object for writing to the LED strip.
APA102<dataPin, clockPin> ledStrip;

// Set the number of LEDs to control.
const uint16_t ledCount = 11;

// Create a buffer for holding the colors (3 bytes per color).
rgb_color colors[ledCount];

// Set the brightness to use (the maximum is 31).
const uint8_t brightness = 31;

void setup()
{
   if (F_CPU == 16000000) clock_prescale_set(clock_div_1);
}

// Converts a color from HSV to RGB.
// h is hue, as a number between 0 and 360.
// s is the saturation, as a number between 0 and 255.
// v is the value, as a number between 0 and 255.
rgb_color hsvToRgb(uint16_t h, uint8_t s, uint8_t v)
{
    uint8_t f = (h % 60) * 255 / 60;
    uint8_t p = (255 - s) * (uint16_t)v / 255;
    uint8_t q = (255 - f * (uint16_t)s / 255) * (uint16_t)v / 255;
    uint8_t t = (255 - (255 - f) * (uint16_t)s / 255) * (uint16_t)v / 255;
    uint8_t r = 0, g = 0, b = 0;
    switch((h / 60) % 6){
        case 0: r = v; g = t; b = p; break;
        case 1: r = q; g = v; b = p; break;
        case 2: r = p; g = v; b = t; break;
        case 3: r = p; g = q; b = v; break;
        case 4: r = t; g = p; b = v; break;
        case 5: r = v; g = p; b = q; break;
    }
    return (rgb_color){r, g, b};
}

void loop()
{
  uint8_t time = millis() >> 4;

  for(uint16_t i = 0; i < ledCount; i++)
  {
    byte x = time - i * 8;
    colors[i] = hsvToRgb((uint32_t)x * 359 / 256, 255, 255);
  }

  ledStrip.write(colors, ledCount, brightness);

  delay(10);
}

Used with the recommended ATTiny bootloader and the 16MHz clock configuration.