What is the fastest and easiest way to reach lvl 3 for the private rank?

Solution 1:

Taking a look the the recent Operation Bloodhound update, where the XP system was launched, it states:

You will earn XP at the end of a match on official servers. In a Competitive match, your earned XP is determined by your rounds won. In all other game modes, your earned XP is determined by your score. Additionally, at the start of each week you will receive an XP bonus boost. During Operation Bloodhound, you can gain extra XP by completing missions.

Ignore the competitive bit, but the fastest way is to play on official server either on missions or a casual mode. The XP generated is based on your score in the matches. So if you have a favorite weapon or map, choose a match mode and map that suit you best.

Matches with instant respawn are your best option to maximize the amount of time you have to get kills, but I think the overall XP gain on a casual match 10v10, if you win with high number of kills, would be more.

Also, I think killing chickens give you 1xp, so kill those pesky pluckers.

Solution 2:

Deathmatch is a pretty good option. You score 600+ points and get a decent amount of xp in 10 minutes. Arms race is also fast, but it depends on you skill level, so you can make it end even faster (i had about 350 points per game).