Hi /u/SamyK - mirrorless cameras are not necessarily better for video - but the GH3 is better than the D5300. Here is why:
the GH3 has an electronic viewfinder that continues working while you're recording video- the D5300's mirror blocks its optical viewfinder when you record video
the GH3 has a headphone jack for monitoring your audio - the D5300 has no headphone jack
The GH3 has an all-metal splash-resistant body - the D5300 has a plastic body with no weathersealing
The GH3 can record continuously for hours - the D5300 shuts down after 30 minutes of continuous recording and has to be restarted.
The GH3 is compatible with inexpensive Panasonic power zoom lenses - there are no power zoom lenses for Nikon DSLRs
If you ever want to use video autofocus, Panasonic makes video optimized lenses with fast, silent autofocus motors - Nikon lenses have slow and noisy video autofocus.
The GH3 (and most mirrorless cameras) can be inexpensively adapted to mount just about any lens - this is much more difficult for Nikon mount cameras.
The GH3 can record at bit rates up to 50mbps IPB and 72mbps ALL-I - the D5300 is limited to 36mbps. Generally (but not always), higher bit rates mean more information recorded to the file and higher quality video.
Here is the image quality you can get from the GH3:
In Camera Slow Motion
I started out with a Nikon D50 (my first DSLR after shooting with film for 30 years), I moved to a Canon T2i, then a GH2, a GH3, then the GH4 - and at $760 body-only, the GH3 is a terrific value for your money.
Hope this is helpful and good luck!