There's also the first 70-200mm f2.8 VR which will be great on the D5300 and cheaper than the VRII. This lens vignettes on FF though so if the OP is considering swapping to FF in the future they may wish to discount it.
AF-S 70-200mm f/2.8G ED VR II Lens £1749.00
AF-S 70-200mm f2.8E FL ED VR Lens £2199.00
AF-S 70-200mm f/4G ED VR Lens £999.00
That is your choice as they don't do a DX version only Full frame
So really it is more a case of your budget and if you intend to upgrade your camera later on to a full frame camera
First question is f2.8 or f4 version
if f2.8 then the first in the list is the one I have
All 70-200 mm lenses in the Nikon range are fairly heavy so would they upset the balance between your camera and lens, just a thought
I don't find it an issue with these lenses as I hold it by the lens anyway.A light one, that body will be very lens heavy otherwise.
Forgot to say as yours is a crop sensor those mm length have to be multiplied by around 1.4 or 1.5 times can't . remember which.
might pay to have a look at the 70-300mm VR lens
AF-S 70-300mm f4.5-5.6 G IF-ED VR Lens £479.00
ok F4.5-5.6 but I used to have that lens until I upgraded and a lot cheaper