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.