You can upgrade them at the personnel officer at Starfleet Academy. However you need to turn in 5 officers of a lower quality as well as dilithium to do so, and it's really not worth it. For instance, Rare Duty Officer Reassignment requires 5 rare officers and 6000 dilithium to give you 1 random very rare officer. This dilithium cost makes it very unprofitable to do these assignments, and is honestly the worst way in the game to get very rare officers.
It's easier to get your very rare officers by doing recruitment missions, opening duty officer packs, and buying them from the exchange. There are also special duty officer assignments that you can do in the exploration zones after you finish the colonization chains that can award very rare officers on a critical success. You can also buy a purple officer from the duty officer store once you reach Tier 4 in any given duty officer specialization. I believe Omega reputation also has assimilated duty officers in it's reputation store.
Uncommon and Rare officers can be traded in by doing the Officer Exchange assignments that give you 3 random lower quality officers - this is good for fueling starbase projects with common officers, and getting rid of unnecessary greens and blues that are taking up space on your roster. If you aren't in a fleet, you can sell off your commons for a decent price too - they are always in demand by players who are in fleets.
*EDIT: Also the first time you score a critical success on "Perform a Holodeck Performance of Hamlet" you always get William Shakespeare, a very rare photonic officer. Any future critical successes on this assignment give you a random purple photonic duty officer.
So unless I'm actively doffing with a character there is a very simple cause of action
white ones : Fleet projects
Green ones: sold in exchange
Blue ones: Sold in exchange if not needed
purple ones: Kept unless I have multiple of the same kind, then they are sold in exchange.
Having only 100 Doff slots (85 Doffs in your case) is simply not enough if you plan on doing Doff missions seriously because you need a good mix qualified Doffs so that you can do a variety of missions to have a very good chance of success. Increasing your various commendations will reward you with a Doff. Commendation level 1 and 2 will allow you to choose a green Doff, commendation level 3 will allow you to choose a blue Doff and finally when reaching commendation level 4 you can choose a purple Doff. There are around 12 different commendations that you can level up.
My Fed toon has a total of 400 Doff slots, my KDF toon has a total of 300 Doff slots and my Romulan toon currently only has the default 100 Doff slots so far. My Fed and KDF toons probably have around 75 - 80 free Doff slots left. Based on my experience you should have a minimum of 200 Doff slots in order to have enough variety of Doffs to be able to do the various Doff missions and level up the commendations.
If you have not started any of the Colony Chain Missions, then that is a good way to get a specific blue Doff as a reward for completing the 7 mission chain to establish a colony in various "Exploration Clusters"; there are a total of 14 such clusters which means 14 blue Doffs.
The last mission in the 7 mission chain is called "Renown" and it rewards a specific blue Doff based on your faction and the cluster which can be found in the following link:
Lastly, once the colony chain has been completed for a cluster it opens up a repeatable mission for a specific purple Doff if you can complete it with critical success. Otherwise you only get a refugee (Fed) or prisoner (KDF). The Fed version is known as "Support Colonization Efforts" while the KDF version is called "Support Expansion Efforts". As stated, these are repeatable missions so you can get multiple copies of the same purple Doff assuming you get a critical success.
If you have a Fed toon, then you are locked out of 4 clusters because they are connected the Klingon home sector. If you have a KDF toon, then you are only locked out of 1 cluster which is connected to the Fed home sector. You can travel into "enemy home sectors" when your Diplomacy (Fed) / Maraudering (KDF) has reached commendation level 3.
This means as a KDF toon you simply cannot get purple Doctors until you reach Maraudering level 3. However, for a Fed toon that has not reach Diplomacy level 3 you cannot get purple Armory Officers, Deflector Officers, Security Officers and Fabrication Engineers.
Comments
It's easier to get your very rare officers by doing recruitment missions, opening duty officer packs, and buying them from the exchange. There are also special duty officer assignments that you can do in the exploration zones after you finish the colonization chains that can award very rare officers on a critical success. You can also buy a purple officer from the duty officer store once you reach Tier 4 in any given duty officer specialization. I believe Omega reputation also has assimilated duty officers in it's reputation store.
Uncommon and Rare officers can be traded in by doing the Officer Exchange assignments that give you 3 random lower quality officers - this is good for fueling starbase projects with common officers, and getting rid of unnecessary greens and blues that are taking up space on your roster. If you aren't in a fleet, you can sell off your commons for a decent price too - they are always in demand by players who are in fleets.
*EDIT: Also the first time you score a critical success on "Perform a Holodeck Performance of Hamlet" you always get William Shakespeare, a very rare photonic officer. Any future critical successes on this assignment give you a random purple photonic duty officer.
So unless I'm actively doffing with a character there is a very simple cause of action
white ones : Fleet projects
Green ones: sold in exchange
Blue ones: Sold in exchange if not needed
purple ones: Kept unless I have multiple of the same kind, then they are sold in exchange.
My Fed toon has a total of 400 Doff slots, my KDF toon has a total of 300 Doff slots and my Romulan toon currently only has the default 100 Doff slots so far. My Fed and KDF toons probably have around 75 - 80 free Doff slots left. Based on my experience you should have a minimum of 200 Doff slots in order to have enough variety of Doffs to be able to do the various Doff missions and level up the commendations.
If you have not started any of the Colony Chain Missions, then that is a good way to get a specific blue Doff as a reward for completing the 7 mission chain to establish a colony in various "Exploration Clusters"; there are a total of 14 such clusters which means 14 blue Doffs.
List of Exploration Clusters:
http://sto.gamepedia.com/Exploration_Cluster
List of missions in the "Colony Chain":
http://sto.gamepedia.com/Assignment_chain:_Colonization
The last mission in the 7 mission chain is called "Renown" and it rewards a specific blue Doff based on your faction and the cluster which can be found in the following link:
http://sto.gamepedia.com/Assignment:_Renown
Lastly, once the colony chain has been completed for a cluster it opens up a repeatable mission for a specific purple Doff if you can complete it with critical success. Otherwise you only get a refugee (Fed) or prisoner (KDF). The Fed version is known as "Support Colonization Efforts" while the KDF version is called "Support Expansion Efforts". As stated, these are repeatable missions so you can get multiple copies of the same purple Doff assuming you get a critical success.
Support Colonization Efforts - Fed
http://sto.gamepedia.com/Assignment:_Support_Colonization_Efforts
Support Colonization Efforts - KDF
http://sto.gamepedia.com/Assignment:_Support_Expansion_Efforts
If you have a Fed toon, then you are locked out of 4 clusters because they are connected the Klingon home sector. If you have a KDF toon, then you are only locked out of 1 cluster which is connected to the Fed home sector. You can travel into "enemy home sectors" when your Diplomacy (Fed) / Maraudering (KDF) has reached commendation level 3.
This means as a KDF toon you simply cannot get purple Doctors until you reach Maraudering level 3. However, for a Fed toon that has not reach Diplomacy level 3 you cannot get purple Armory Officers, Deflector Officers, Security Officers and Fabrication Engineers.