I would say this item is rare due to the "free" extra attack
Regarding the +5 to initiative part
The obvious comparison is the Alert feat, which gives the following benefits:
- You gain a +5 bonus to initiative.
- You can't be surprised while you are conscious.
- Other creatures don't gain advantage on attack rolls against you as a result of being unseen by you.
This is something a player character could have a level 4 (or level 1 if they're a variant human), so this implies that it isn't unbalanced for a player character to have an extra +5 to initiative at level 5.
If your item is roughly equivalent to a third of a feat (or half a feat if we assume that the +5 is the strongest part of this feat, which I'm not entirely sure is true), then finding magic items that are roughly equivalent to half a feat would be a good way to measure the rarity of this amulet.
Using the Alert feat again, the weapon of warning, an uncommon magic item that requires attunement, says:
While the weapon is on your person, you have advantage on initiative rolls. In addition, you and any of your companions within 30 feet of you can't be surprised, except when incapacitated by something other than nonmagical sleep. The weapon magically awakens you and your companions within range if any of you are sleeping naturally when combat begins.
This also seems to be roughly equivalent to half a feat, since it does everything the Alert feat does (if we consider +5 and advantage to be equivalent, as the rules for passive checks imply) except for cancelling out advantage for unseen enemies. If this is considered an uncommon item, I don't see how this aspect of your item would be considered rarer. However, it is clearly more powerful than the minor benefits of a common item, so I'd say that puts it quite squarely in uncommon territory.
As for using proficiency instead of the flat +5, I assume this suggestion was an attempt to make it less powerful if +5 somehow made this a rare item. Firstly, the fact that there's nothing official that grants proficiency to initiate, and that very few things add your proficiency bonus as a modifier to things outside of attack rolls, skill checks, saving throws and spell bonuses, implies that it probably shouldn't be done. What's more, it's actively worse before level 13, the same as a flat +5 until level 17, at which point they only have 1 more than your +5 bonus anyway. So I'd say keep away from giving proficiency to initiative and just keep the flat +5.
One final thing to be wary of; do you intent for this to stack with the Alert feat? Having a +10 to initiative with little effort might not be what you were intending. You could put in the magic item's description that it doesn't stack with the Alert feat, although that isn't something I've seen before on any official magic item (i.e. "this magic item only works if you don't have X feat"; seems odd to me). Maybe as an attunement requirement?
Regarding the additional attack part
This is essentially the same as the Gloom Stalker ranger's ability, Dread Ambusher, the 3rd level feature, which says (XGtE, p. 42):
If you take the Attack action on that turn, you can make one additional weapon attack as part of that action. If that attack hits, the target takes an extra 1d8 damage of that weapon's damage type.
Your item doesn't give the extra 1d8 damage, but it does allow the extra attack. Furthermore, this doesn't use up their bonus action, like a scimitar of speed:
You gain a +2 bonus to attack and damage rolls made with this magic weapon. In addition, you can make one attack with it as a bonus action on each of your turns.
So, again, yours doesn't have the +2, just the extra attack part, but without using a bonus action. I think if it did use a bonus action, then I'd say that plus the +5 to initiative is probably still uncommon, but giving them an extra attack without costing them anything regarding their action economy pushes this into rare, since you're essentially handing out Extra Attack for free (to classes that don't have it, and Extra Attack (2), the Fighter's 11th level version, to classes that do have it, such as a ranger) for that first round of combat.
The reason I rate this highly, despite only being one round, is that, unlike the scimitar of speed (besides the bonus action thing mentioned above), this isn't limited to a specific weapon (i.e. a scimitar), but rather any weapon you are holding; this could apply to other magic items too. The Dread Ambusher feature does the same, so it's not unprecedented, but handing out a subclasses' level 3 feature should not be done lightly, especially since (you mentioned in comments that this is intended for a Hunter ranger) this ranger now has the features of two ranger subclasses (almost) thanks to a magic item.
If this were to use your bonus action, and still be restricted to first round only, then that may push it back down to uncommon, but allowing a bonus action attack every turn is then rare again due to this being an upgrade of the scimitar of speed, then +5 to initiative on top of that.