For starters, you want to write brother in ancient greek.

1. Type adelfos to get an understanding of what’s happening.
   
2. Set the input method to greek-babel: M-x set-input-method greek-babel.¹
   
3. Type adelfos again.
   

The last typing of “adelfos” results in αδελφοσ. So far so good.

However, αδελφοσ is wrong. It should be ἀδελφός. To be specific, we’re missing diacritics on a (alpha) and o (omicron) and the final s (sigma) is wrongly written as well.

We solve the problem by using combinations of letters, such as:
Typing >a results in ἀ
Typing 'o results in ό
Typing c results in ς

For completeness, type >adelf'oc in order to eventually get the desired ἀδελφός.

Good 🙂

Go back to your previous input-method via M-x toggle-input-method

Documentation

Read the documentation M-x describe-input-method greek-babel

Input method: greek-babel (‘BG’ in mode line) for Greek
  The TeX Babel input method for Classical Greek using mule-unicode-0100-24ff.

-------------------------------------
character     capital	      small
-------------------------------------
alpha		A		a
beta		B		b
gamma		G		g
delta		D		d
epsilon		E		e
zeta		Z		z
eta		H		h
theta		J		j
iota		I		i
kappa		K		k
lambda		L		l
mu		M		m
nu		N		n
xi		X		x
omicron		O		o
pi		P		p
rho		R		r
sigma		S		s
final sigma			c
tau		T		t
upsilon		U		u
phi		F		f
chi		Q		q
psi		Y		y
omega		W		w
-------------------------------------
sampi				!
digamma		#
stigma				$
koppa		&		%
-------------------------------------

------------------------
mark		key
------------------------
ypogegrammeni	|
psili		>
dasia		<
oxia		\='
koronis		\='\='
varia		\=`
perispomeni	~
dialytika	"
ano teleia	;
erotimatiko	?
----------------------

Optional - Increase readability

To increase the readability –of the in particular tiny diacritics–, you can enlarge the greek characters by doing the following:
Install the Gentium font: $ yay ttf-gentium
Evaluate this line or add it appropriately to your init.el:

(set-fontset-font "fontset-default" 'greek (font-spec :family "Gentium" :size 30))

Sources

https://stackoverflow.com/a/49765665/1172907
https://youtu.be/Fl4eu_bv54c?t=266