To get a truly convincing effect, you would have to start more or less from scratch: build up the phoenix in layers by drawing it with cloned parts of fire. You can use the existing graphic as a template to do this, and remove it later.
https://design.tutsplus.com/tutorials/how-to-create-a-hellacious-flaming-skull-in-photoshop--psd-2894
Download fire brushes to make things easier: Fire Free Brushes - (581 Free Downloads)
Another option is to burn a real wooden phoenix model and composite that into your image, of course (would still need editing). And the background would need to be keyed - requiring a black background, for example.
Or use a 3d app like Blender to create a 3d Phoenix that is actually "on fire", and composite it in your image. Might seem like a lot of work, but results in very realistic looking effects.
Blender Phönix Test - YouTube
Quick Fire & Smoke Setup in Blender 2.7 Cycles - YouTube
How to Render Fire in Cycles - YouTube