Egill Skallagrímsson comes to mind immediately, hero of one of the Viking sagas. Egil is one of those semi-mythical figures who really did exist, but likely not in exactly the way he was depicted. If we go by the 13th-century Egil's Saga, he was a giant of a man, deft in swordplay, devious in plans, and well-versed in the art of Odin's mead, Skaldic poetry and verse.

He was also skilled in the art of runes and the scorn-pole, according to legend: after being wronged by the king of Norway, he erected a scorn-pole on the eastern shores of Iceland, a pole mounted with a horse's head and carved with runes of cursing, sending spirits against his Norwegian foe. In an earlier encounter, he engraved runes on a cup to check for poison, and the cup split in half, spilling the poison on the floor.

Obviously this should all be taken with a grain of salt, but Egil was likely a superb warrior who had the additional gift of skaldic verse, which he frequently demonstrated to various kings and nobles. Skaldic verse was extremely complex, using a syllabic-construction for the verses in which every syllable of every line was carefully counted. It also used "kennings", which could best be described by showing it.

If I wanted to narrate a poem about the great battle-feats of my king, Harald Hard-Ruler, I would couch each line of the poem in elaborate imagery. For example, if I wanted to describe Harald as mighty in battle, I might say:

"Then did the ring-giver redden the wolf's tongue, as through the battle-shower he strode"

or

"Harald (ring-giver), killed people (reddended) with his sword (wolf's tongue) while dodging the arrows (battle-shower).

Wikipedia helpfully actually has one of Egil's poems:

Þat mælti mín móðir,

at mér skyldi kaupa

fley ok fagrar árar,

fara á brott með víkingum,

standa upp í stafni,

stýra dýrum knerri,

halda svá til hafnar

höggva mann ok annan.

"My mother wants a price paid

To purchase my proud-oared ship

Standing high in the stern

I'll scour for plunder.

The stout Viking steersman

Of this shining vessel:

Then home to harbour

After hewing down a man or two."

I think Egil would count as a warrior-poet. I don't have hard evidence for the term "warrior poet" coming from Viking sources, but I think it certainly contributed; many of the Norwegian kings were depicted in the sagas as gleefully throwing out poems and songs as they fought, disemboweling enemies while dropping hilarious kennings about it.

There's also stories of battle with words in Icelandic literature, known as flyting. In the example I can think of, a Christian missionary named Thangbrand meets an Icelandic priestess called Steinnun, who beats the tar out of him in a word-battle:

'Have you heard that', said she, 'Þórr challenged Christ to a duel, and he dared not fight with Þórr?'

'I have heard', said Thangbrand, 'that Þórr was nothing but earth and ashes, if God wished him not to live.'

She said, 'Who broke the ship you had?'

'What say you on that?' said he. 'That shall I tell you here', said she.

Thor drove Thangbrand’s ship from her moorings

The Powers broke the sea-bird of the bell-ward

Christ did not guard the vessel

God did not watch over the boat.

Thangbrand had no reply to Steinnun's verbal smackdown, so he lost the flyting. The kennings in Steinnun's reply are quite fun too; the Sea-Bird (boat) of the Bell-Ward (missionary), etc.

The samurai of feudal Japan were often poets. Some early examples are found in records of the Gempei wars (1180-1185), where two of the most powerful samurai clans of their time battled for control over the imperial throne. Taira Tadanori, brother of the head of the Taira clan, wrote a few poems that appear in the anthology Senzaishu ("The Collection of a Thousand Years"). Lower ranked samurai, such as Taira Yasunori, were also known to compose poems.

In the late 1500's, there was a warlord named Oda Nobunaga who was betrayed by a trusted general of his, Akechi Mitsuhide. Mitsuhide often collaborated with well-known poets, and even expressed his secret desires to rebel against his lord in a renga (a collaborative form of poetry).

Sources:

The Tales of the Heike, Translated by Burton Watson; Edited by Haruo Shirane

Senzaishu, Fujiwara no Shunzei

It's not an exact parallel with the title "Warrior Poet", but there are numerous depictions in classical Greek and Roman literature of warriors who are also talented in storytelling and music, particularly in epic. Achilles plays the lyre and sings to his guests in Book 9 of the Iliad, and the entirety of Books 2 and 3 of the Aeneid comprise Aeneas' recounting to Dido the "unspeakable grief" of the Fall of Troy and his subsequent wanderings throughout the Mediterranean.

The concept of the warrior poet is one that is common to many societies. Because it is found in societies with no contact with each other, it must have developed independently in different societies. Other commenters have already discussed the warrior poet in Europe, so here's some info about the warrior poet in Aztec society, just to show how the concept is something that really did develop all over the place. In Mesoamerica the concept of the warrior poet was well established. Many Aztec warriors would write poetry when they were not off fighting battles (I remember reading in one of my books that being a poet was the greatest glory for a warrior during the part of the year when they were away from battle), and much of Aztec poetry describes the glories of war. In fact, dying in war was often in poetry called the "flowery death", and poetry itself was called "flowers and song". A ruler of the city of Texcoco named Nezahualcoyotl served as a general and wrote many poems. It is unknown how many of the poems attributed to him he actually wrote (there are many) but even if he wrote none of them, it shows that the concept of a ruler who fought and wrote poetry existed and was well respected, for Nezahualcoyotl was called by some the "best ruler to ever rule in Mexico". Anyway, the warrior poet does not have just one point of origin. Hope this answer helps!

It's basically a relatively common phrase, not something with a specific coinage that you're going to be able to trace back to one originator, but it still has a history. In a nutshell, I'd say, it's attributable to the hunger for mythic founding figures brought on by the rise of Romantic nationalism in the nineteenth century. Here is an Ngram view of the phrase. You'll see that the phrase is essentially unknown (to Google's scans, at least) until the late 18th Century and then picks up significantly in the mid-19th Century. If you click through a few pages of 19C Google Books searches you'll see it being applied to beloved historical figures — Greek, Biblical, medieval, Celtic, etc — and also to contemporary Romantic-nationalist heroes like Kossuth and Kosciusko, but the use of the phrase is clearly a product of the 19C's new hunger for Romantic national founder-figures who were at once statesmen/soldiers and artists, pragmatists and artists.