You can and no adapter is needed (so long as it is an f-mount lens); but I don't think you'd want to.
The lens is not very good optically. On the d5300 it would be manual focus only.
If the lens is not an f-mount, it won't work at all. No adapter would work to any reasonable degree of satisfaction. The lens was available in several mounts one of which was the Nikon f-mount.
If you want a cheap telephoto, I would instead suggest the dirt cheap (and very light weight) Nikon 55-200mm VR, it can be had on Amazon for around 80. (which is less than the price of most adapters)