Definitely don’t need to look over your shoulder or signal, and if you do then by the time you look forward again to make your move then the gap you had your eye on will almost certainly have gone!
Moving up (and just as importantly staying there once you’ve got there) is about having constant positive intent and alertness to grab every inch that you can and then not let it go. Contact is frowned upon - though it happens - but if somebody leaves a big enough gap for you to get your bars in there in front of them and then inch your way across then that’s generally fair game though you’ll get shouted at occasionally. You should also avoid big swerves or braking, that sort of behaviour tends to get you marked out as a liability.
Personally I’ve never found that moving up through a pack comes very naturally however many races I do, it’s something I always have to force myself to do. I’m a confident enough handler and do plenty of fast group riding, so I have no issue cornering in close proximity to others. I’m just not a particularly confrontational person - I’m also a Brit, as a nation we have a fairly strongly engrained affinity for queueing and people waiting their turn, moving up always feels to me a bit like pushing in on a queue! I’m better at maintaining my place, since that queueing mentality works in my favour and I will quite happily deploy every passive aggressive technique in the book to stop other people from “queue jumping” in front of me. If the course allows, then my preference is therefore to look for opportunities to move up the pack without having to fight for every inch - e.g. braking later into a corner, or going wide on a corner and carrying more speed round the outside then cutting back in while the pack is stretched out and there are plenty of gaps as everybody gets back up to speed. Or looking out for riders who are more aggressive than me and jumping on their wheel as they move up - northern Europeans are great for this, the Dutch and Belgians seem to grow up doing elbow to elbow racing on technical courses! I also tend to avoid courses that I know don’t suit me - events with large fields and fast narrow courses where there’s minimal opportunity to move up round the outside aren’t really my thing. I much prefer either a wide course where there is space to make some moves, or one with enough climbing and corners to break the pack up and give me gaps to move into. Though narrow with climbs can also be a problem as if you’re not near the front you can easily find yourself blocked in when a gap forms or a break goes on a climb.