Everyone can rank-up just by killing someone of a higher rank than you. Other than that here are the requirements.
Shinigami:
To rank up as a shinigami there are only 3 ways. One is with the support of one other admin I can rank you up from you completing missions and proving yourself to me. The second way is i will be hosting a tournament, Each rank gets its own division in the tourney. The ones that prove they have skill will get to rank-up. There will be no killing in this tournament you have to get them into a position where you have thier lives in your hands with no dispute. The last way is ONLY to get promoted to captain rank. You challenge a captain and you have to kill him. (will be nearly impossible unless you're damn good so your best bet is the other two.)
Vizored:
Vizored have only one way. Kill those stronger than you. If you want to rank up in Vizored, I suggest achieving Captain rank with the Soul Society before joining the Vizoreds.
Arrancar:
Hueco Mundo has fallen under the rule of Angron Grim, a legendary Arrancar in his own right. Since then he has taken the time to recruit only special Hollows that he finds of interest, in raw energy, power, or intellect. What catches his watchful gaze or the gaze of his leading espada can be sure to have a difficult but rewarding road.
Those that are still Hollow will have a very long road ahead of them. They must defeat two mortals, or their equivalent in soul energy. (One Shinigami, Quincy, or Fullbringer.)
Once this task has been carried out, and it should be rather simple. They will advance to Adjuhas.
Upon reaching this stage, they must prove they have something that is worth having in the upper-echelons of the Arrancar. If they can convince an espada they are worth while, a test will be placed before them to aid in becoming a Vasto Lordes. This test comes in many different ways according to the skills of the Adjuhas.
Once a Vasto Lordes they near the chance for true power. Something to make many within the mortal and soul realms quake. The power of an Arrancar. At this stage, they have proven worthy enough to join the ranks of Angron's Army. They must now show not only power, but a willingness to do what is necessary. They must prove their allegiance beyond a shadow of a doubt. These tests are administered only by the top level Espada. These individuals know the secret to unlocking the abilities of an Arrancar. However it is not something they can hand out at random. Without the Hogyoku it takes a lot of energy on their part.
This task will often involve having an entourage of your own amongst the hollows in Hueco Mundo. You and your force must challenge and defeat either a shinigami or quincy group. Or contribute to the intelligence of the Espada in some way. (Defeating an actual rper will get you there. Defeating nameless shinigami or Quincy will require a 1000 word post.)
In order to obtain the final goal of all Hollows, one must fight one of the current Espada and be deemed worthy by Angron. This will be a true test of skill. It is the choice of Angron who fights whom to make it as difficult as possible. Making an Espada requires intricate work that only Angron has figured out since Aizen's own experiments nearly a hundred years ago. Angron will stop a fight before either side dies, but serious wounds will occur. If you are deemed worthy, Angron will unlock you Espada form and Ressurecion. At which point a 1500 word post is required, to describe your transformation, and ressurecion.
After this your place among the Espada will be given. You will only rank up once per week. If you fail to advance you may try again one time in the same week.