I don't think I've seen a single distro that ships Korean support right off the bat, except HarmoniKR. Because of the way Korean works, it's not just the keyboard layout support you need. To list the things you need to do:
- Install Fcitx5.
- Install Korean Fcitx5 input method. Probably named something like
fcitx-hangul. Without it, Hangul keyboard will not appear in Fcitx. - Check what 한/영 key really does. This is necessary because depending on the hardware, it either emits Hangul key, or Right Alt. If it emits Hangul, then you can skip this. If not, make sure you remap Right Alt to Hangul, because then the key triggers Alt action every time language is swapped. Those usually steal input, so it is really annoying.
- Add both Hangul keyboard (not Korean) and English keyboard in Fcitx5 config. Korean keyboard does not enter Korean characters. It's the Hangul keyboard that actually enters Korean characters. It probably corresponds to Korean keyboard's English input. It should look something like this.

- Add Hangul key as Enumerate Input Method Forward option in Fcitx config under Global Options tab.
I will attach my files in ~/.config/fcitx5. These two should go under that directory, and not ~/.config/fcitx.
~/.config/fcitx5/config
[Hotkey]
# Trigger Input Method
TriggerKeys=
# Enumerate when press trigger key repeatedly
EnumerateWithTriggerKeys=True
# Temporally switch between first and current Input Method
AltTriggerKeys=
# Enumerate Input Method Backward
EnumerateBackwardKeys=
# Skip first input method while enumerating
EnumerateSkipFirst=False
# Enumerate Input Method Group Backward
EnumerateGroupBackwardKeys=
# Time limit in milliseconds for triggering modifier key shortcuts
ModifierOnlyKeyTimeout=250
[Hotkey/EnumerateForwardKeys]
0=Hangul
[Hotkey/EnumerateGroupForwardKeys]
0=Control+Alt+Hangul
[Hotkey/ActivateKeys]
0=Hangul_Hanja
[Hotkey/DeactivateKeys]
0=Hangul_Romaja
[Hotkey/PrevPage]
0=Up
[Hotkey/NextPage]
0=Down
[Hotkey/PrevCandidate]
0=Shift+Tab
[Hotkey/NextCandidate]
0=Tab
[Hotkey/TogglePreedit]
0=Control+Alt+P
[Behavior]
# Active By Default
ActiveByDefault=False
# Reset state on Focus In
resetStateWhenFocusIn=No
# Share Input State
ShareInputState=No
# Show preedit in application
PreeditEnabledByDefault=True
# Show Input Method Information when switch input method
ShowInputMethodInformation=True
# Show Input Method Information when changing focus
showInputMethodInformationWhenFocusIn=True
# Show compact input method information
CompactInputMethodInformation=True
# Show first input method information
ShowFirstInputMethodInformation=True
# Default page size
DefaultPageSize=5
# Override Xkb Option
OverrideXkbOption=False
# Custom Xkb Option
CustomXkbOption=
# Force Enabled Addons
EnabledAddons=
# Force Disabled Addons
DisabledAddons=
# Preload input method to be used by default
PreloadInputMethod=True
# Allow input method in the password field
AllowInputMethodForPassword=False
# Show preedit text when typing password
ShowPreeditForPassword=False
# Interval of saving user data in minutes
AutoSavePeriod=30
It contains Japanese group, you can safely delete that. I just don't know what it looks like if I delete it.~/.config/fcitx5/profile
[Groups/0]
# Group Name
Name=日本語
# Layout
Default Layout=us
# Default Input Method
DefaultIM=mozc
[Groups/0/Items/0]
# Name
Name=keyboard-us
# Layout
Layout=
[Groups/0/Items/1]
# Name
Name=mozc
# Layout
Layout=
[Groups/1]
# Group Name
Name=한국어
# Layout
Default Layout=us
# Default Input Method
DefaultIM=hangul
[Groups/1/Items/0]
# Name
Name=keyboard-us
# Layout
Layout=
[Groups/1/Items/1]
# Name
Name=hangul
# Layout
Layout=
[GroupOrder]
0=한국어
1=日本語
