The simple answer is, no, you can't do it.
The longer answer is, yes, you can do it, but it will take more than you think and will probably be harder.
Let's take a group of 8 LEDs. You can drive one from each output of a 595, right? (As long as you don't try to drive them too hard).
Now you want to drive 3 inputs to each LED. This will take 3 595s - call them the red595, green595 and blue595. With this setup, you can only turn each LED on or off, so you get 7 different colors (8 if you count black as a color).
Now you want to turn each LED partially on or off, and each one independently. You can do this by providing each 595 with PWM data. So lets look at LED1, the red channel. Let's say you want to control its intensity in 8 steps. What you can do is update it in cycles of 8 updates. Let's say you want to set it to level 3 on a scale of 0 to 7. The binary equivalent of 3 (in octal representation) is 011. So you would send a cycle of red values that look like 1,1,1,0,0,0,0,0. The LED would be on for 3/8ths of a cycle, at full brightness when it is on.
Now, lets say you have 100 LEDs. It will take 13 595s to control them. The LEDs must have a PWM frequency better than about 30-50 Hz, or they will appear to flicker, and higher is better for colored LEDs. Let's say you run at 60 Hz. To update the entire string of LEDs will take 100 clock cycles. Since each PWM takes 8 cycles, the 595s must be fed at a clock rate of at least 100 x 8 x 60, or 48,000 Hz. For each data output, the computer must determine which pixel is being sent, what its value is, and what the PWM value should be. That's not hard at 48 kHz. For 3 colors, the effective clock rate must be 3 times 48 KHz, or 144 kHz. Still not too bad (probably). The physical clock rate can remain at 48 kHz, since you can use 3 lines for data, set them up, then send a single clock to all 3 x 13, or 39 595s.
But let's say that you want 256 intensity levels. Now the clock rate has to be 256 x 100 x 60, or 1.53 MHz, and that's starting to look rather iffy. Not only that, you can still use this physical clock rate for 3 colors, since you can use 3 data lines and clock the 595s from the same clock. But. For each physical clock you have to determine 3 separate color values, calculate 3 different PWM values, and output those 3 bits to data lines. And all in about 650 nsec.
You may find this something of a challenge.