You can upgrade a weapons' level by using the exp orbs (red ones) you collect from doing missions throughout the game. Simply go onto settings, raise stats, weapons, and select the weapon you desire to upgrade.
To get it to level 10 however takes a fair amount of exp, and without the soul drain skill it'll likely take you the majority of the campaign.
If my calculations serve me correctly, you will need a total of 90000 exp to get a weapon to level 10.