This is my guide to all the character portraits, as far as I know you need level 5 exploration for your Dog, I didn't find any until I had it so I assume that is true.
Elise Portrait - This item is given to you randomly as a gift when you marry Elise as a male character.
Elliot Portrait - Like Elise, this is given randomly as a gift when you marry Elliot as a female character.
Sabine Portrait - This one is found in a dig spot in Dweller's Camp next to the caravan where you meet with Sabine.
Saker Portrait - This portrait is found in a dig spot in the Mercenary Camp behind the first buy-able house you encounter when you enter the camp.
Logan Portrait - This portrait can be found in a dig spot in the Castle Garden to the right of the Catacombs entrance in front of the Logan Statue (not the one in the center, the one on the right).
Walter Portrait - This one can also be found in a dig spot in the Castle Garden, in the far back on the left, in the section to the left of the Walter statue.
Major Swift - This portrait can be obtained by randomly by earning a score above 2000 in the Mourningwood Fort Mortar Game after you receive the Legendary Weapon.
Ben Finn - This portrait is found the same way as Major Swift, by earning a score above 2000 in the Mourningwood Fort Mortar Game after you receive the Legendary Weapon.
Theresa - This portrait is found in a dig spot in the old Gypsy Camp, you must have constructed the bridge to get to Driftwood in order to get to this one, cross the bridge and just before the entrance to Driftwood you should be able to find this one easily.
Page - This one is trickier to find, it's in the Revolution Headquarters under Bowerstone Industrial, You can't get to it until you have gained Page as an ally and are on your way with Ben Finn to the docks, head out of the door that is opened and cross a wooden bridge, after the bridge there will be a large pile of rubble to your right(you know your in the right spot because there will be sparks above the pile) at the back of the pile is a small path leading to the section of the sewers with a chest, you can't miss it.
Reaver - This portrait can only be obtained with the Understone Add-On, it is a reward for ranking up with the Wheel of Misfortune.
Kalin - This one I only know of as a rumor, if anyone has found it or knows for sure it does not exist please tell me.
I have seen reports of people getting these from other areas, such as Elise or Elliot from digging in the Castle Gardens, and Ben Finn from the graves/graveyard in the Mourningwood Fort, I have been unable to confirm these myself but I don't doubt the people who gave me the reports, all the locations I have listed are ones I have confirmed myself multiple times. If anyone knows of others I may have missed or dont know of let me know.
Elise Portrait - This item is given to you randomly as a gift when you marry Elise as a male character.
Elliot Portrait - Like Elise, this is given randomly as a gift when you marry Elliot as a female character.
Sabine Portrait - This one is found in a dig spot in Dweller's Camp next to the caravan where you meet with Sabine.
Saker Portrait - This portrait is found in a dig spot in the Mercenary Camp behind the first buy-able house you encounter when you enter the camp.
Logan Portrait - This portrait can be found in a dig spot in the Castle Garden to the right of the Catacombs entrance in front of the Logan Statue (not the one in the center, the one on the right).
Walter Portrait - This one can also be found in a dig spot in the Castle Garden, in the far back on the left, in the section to the left of the Walter statue.
Major Swift - This portrait can be obtained by randomly by earning a score above 2000 in the Mourningwood Fort Mortar Game after you receive the Legendary Weapon.
Ben Finn - This portrait is found the same way as Major Swift, by earning a score above 2000 in the Mourningwood Fort Mortar Game after you receive the Legendary Weapon.
Theresa - This portrait is found in a dig spot in the old Gypsy Camp, you must have constructed the bridge to get to Driftwood in order to get to this one, cross the bridge and just before the entrance to Driftwood you should be able to find this one easily.
Page - This one is trickier to find, it's in the Revolution Headquarters under Bowerstone Industrial, You can't get to it until you have gained Page as an ally and are on your way with Ben Finn to the docks, head out of the door that is opened and cross a wooden bridge, after the bridge there will be a large pile of rubble to your right(you know your in the right spot because there will be sparks above the pile) at the back of the pile is a small path leading to the section of the sewers with a chest, you can't miss it.
Reaver - This portrait can only be obtained with the Understone Add-On, it is a reward for ranking up with the Wheel of Misfortune.
Kalin - This one I only know of as a rumor, if anyone has found it or knows for sure it does not exist please tell me.
I have seen reports of people getting these from other areas, such as Elise or Elliot from digging in the Castle Gardens, and Ben Finn from the graves/graveyard in the Mourningwood Fort, I have been unable to confirm these myself but I don't doubt the people who gave me the reports, all the locations I have listed are ones I have confirmed myself multiple times. If anyone knows of others I may have missed or dont know of let me know.