From the main menu select Play and then select Play Online, Using R1 and R2 select Competitive.
From here you must complete 10 matches in one match type (3v3 Standard / 2v2 doubles / 1v1 Solo Duel / 3v3 Solo Standard) So for example completing 5 matches in 1v1 Solo Duels and then completing 5 matches in 3v3 standard will not count.
There are different criteria for unlocking this based prior gameplay, so I will list them separately below:
If you are already currently placed in a match type: You can play a single match in that match type and upon match completion the trophy will pop.
If you have completed part of placement matches in a game type (i.e.: 6/10 matches in 1v1 Solo Duel): You only need to complete the rest of the games to 10, so if you have 5 done, just do 5 more, and if you had 6 done, just do 4 more, etc. Upon finishing these matches the trophy will pop.
If you have not done any placement matches in a game type: You will need to pick a game type and complete 10 matches in that game type and the trophy will pop.