This is supposed to be supplementary to Man785's solution.
Also I will be using 1-10 for the gun levels rather than the roman numerals. If you have problems converting between the two, that's a you problem.
First and foremost, even if you never use the M-8 Avenger upgrade it to level 5 before finishing your first playthrough. I will explain why below in section 3.
1. The NG+ Bump
All weapons obtained in your first playthrough get bumped up a level on your second playthrough (new game +). So if you upgrade any weapon to level 5 it will be at 6 upon starting the second playthrough. I had 9 guns all at level 5. They all were bumped to level 6 and everything that I left at level 1 was at 2 upon starting the second playthrough.
2. Reacquire Bump
During your second playthrough weapons get bumped up a few levels when reacquired. You get +2 upon reacquire if the weapon was at level 5 when you finished the first playthrough. You get +3 upon reacquire if it was below level 5.
3. M-8 Avenger
So, why did I ask you to upgrade the M-8 Avenger? Simple. It is the first gun you reacquire in your second playthrough.
Towards the end of the prologue with Anderson you can grab it off the ground by the beacon used to signal the Normandy. The pistol Anderson gives you does not count as a reacquire.
Assuming you upgraded the M-8 Avenger to level 5, it will be instantly bumped to level 6 for New Game + and then bumped to level 8 when reacquired in the prologue.
After that you only need 17500 credits for the last two levels (7500 for level 9 and 10000 for level 10) to upgrade when back on the Normandy.