To get all of the normal belt achievements it requires you to play online in the quick match mode, this is considered the ranked play online. After the match you will gain experience which will go towards ranking up, if you win you get 30 xp automatically and if you lose 10. You can also gain experience though bonuses rewarded based on how you fight but these don't count for much xp however, generally 2 to 4 xp each. The highest xp bonus I have seen for a fight is an extra 10 xp for winning or defending a belt title, aside from those it is either the 2-4 bonus xp or nothing. The nice thing however is your total xp is multiplied by how many rounds you go before the match is over, so if you go 2 rounds you get twice the xp, three you get three times, ect ect.
This is still a time consuming process unfortunately. When you rank up, even if you are a white belt, you get a stripe. You have to get 4 stripes to get the next belt color, so in other words you need to rank up 5 times before you get the next belt. The fortunate thing however is the amount of xp DOES NOT CHANGE until you get the next belt.
The best way to go about ranking up in the least amount of time possible is to get on with a friend during non peak hours, aka early morning, and hope that you get matched up. To increase the odds of getting matched up after you pick your weight class and fighter wait until the other player has done the same and start searching at the same time. Most times a couple seconds after doing this you are matched with them, and from there you can discuss how you are going to play out the fight if you are.