How can I upgrade my weapons to level 10?

To get to level 10 with a weapon, you need to complete the game then import your character on a new playthrough, where you will be able to buy/collect the remaining upgrades.

You can upgrade weapons to level 7 at the start of new game plus and you will get 3 automatic levels upon collecting the weapon again in the next playthrough.

Your other option is to buy Recruit packs in multiplayer. They will sometimes contain weapon upgrades and you can get up to level 10 that way.

Each pack costs 5000 credits and the contents are random, but it might be easier than doing a double single-player playthrough.

For others arriving at this looking to upgrade without completing the game first:

Mass Effect 3 Gibbed save editor. Go to raw, player, change new game count from 0 to 1. Weapon X upgrades unlocked.

He later posted a correction which I overlooked:

My fault, was typing that advice out of memory. It's actually "raw/plot/plot". To change weapon mods past level 7 without finding the weapon itself go to "raw / plot / player variables" then just look up the weapons you want to change, to find the items easier just focus on the numbers with 7.

Credits go to LeonZweihander on gamefaqs.com