How to interface PWM output from a microcontroller with power MOSFET?
That's a rather overkill FET for the purpose.  For lower current applications, and when you get a FET rated for lower voltage, which then has lower overall gate charge, you can sometimes drive the FET gate directly from a digital output.

However, in this case, you should use a FET driver.  For example, the Microchip TC442x is a nice low side FET driver, although that particular one is dual channel.  FET drivers take a digital signal as input and have output drivers specifically designed to provide the large pulses of current it takes to switch a FET gate quickly.  Especially if this is a one-off project, just do it.

If this is a high volume project where cost is a high priority, I'd look around carefully to find a FET that just meets the requirements and has low enough gate charge and works well enough with 5 V gate drive to run directly from a dsPIC output.  I have done that quite successfully a number of times, but 5 A is getting to where the tradeoffs may go the other way.  Another possibility is to use two such direct-drive FETs and have each drive 2 banks of LEDs.  Switching 2.5 A twice will be easier than switching 5 A once, especially if you can use two PWM outputs that are 1/2 period offset from each other.