The trick to getting this solo is to use your 1st finger in a positional way. Notice in the ActionTab how the red dot is often switching between different fret positions during play.

Although things start slowly enough there are some difficult parts to the solo, so be patient when practicing. For the fast parts, just isolate them and practice them individually, slowly first, then build speed.

Blackmore uses the whammy bar a lot for adding vibrato to bends. In this solo he does it here, here, here, here and at the end. He's also not adverse to the occasional divebomb either!

You don't need a whammy bar to do those vibrato effects. You can waver the string whilst the string is at the apex of its bend using your fingers. It's a bit more awkward, but the only way on guitars with no whammy system. As for the divebomb, you can just let the G sound instead (it works fine). Or, if you're keen, go to fret 3 on the D string (or fret 8 on the Low A) and bend it up until it sounds like a G. Then pick it and slowly release / return the note.

Lastly, there is a nice long bend that he picks repeatedly. Use your ear to get a feel for that one. Bend the note up quickly and pick it whilst in full bend, then release the bend slowly, all the time picking it.