No biographical details are currently known.
I identified him through his credited role in Torrejón City, then found him in most of the 50's and 60's roles from the 31 titles listed by IMDb, not one of which matched with the 25 titles where we already had him identified! While he has a rather high number of credited roles for a background actor, many of these credits actually turn out to be in minuscule roles that you would not expect to have been credited.
His IMDb filmography includes 4 Argentine films from the 1940's and 50's. I would guess at these being another actor of the same name, particularly since we have now found Spanish roles for Enrique in these decades.
- Tarjeta de visita .... Head waiter (uncredited)
1947 - Alma canaria .... (credited, but not found)
1948 - Aventuras de don Juan de Mairena .... Poker player (uncredited)
1949 - Filigrana .... Man with the Count at the recital (credited)
1955 - El guardián del paraíso .... (credited, but not found)
- Historias de la radio .... Radio station receptionist (uncredited)
- La lupa .... Nightclub customer (uncredited)
- Suspiros de Triana .... Barman on the ship (uncredited)
1956 - El fenómeno .... Man at the conference (uncredited)
- Esa voz es una mina .... Waiter / Singing contest audience member (uncredited)
- La fierecilla domada .... Servant of Beltran (uncredited)
- La vida en un bloc .... Nightclub customer (uncredited)
- Manolo guardia urbano .... Man at the christening (uncredited)
- Tarde de toros .... Bullfight spectator (uncredited)
1957 - Faustina .... Nightclub customer (uncredited)
- La Cenicienta y Ernesto .... Guest at the ball (uncredited)
- Las muchachas de azul .... Waiter (credited)
- Susana y yo .... Member of school staff (uncredited)
- Totò, Vittorio e la dottoressa .... Wedding guest (uncredited)
- Un marido de ida y vuelta .... Board member (uncredited)
1958 - Aquellos tiempos del cuplé .... Member of parliament (uncredited)
- El Cristo de los Faroles .... Man backstage with Relámpago (credited)
- El fotogénico .... Dance contest spectator (uncredited)
- El hincha .... Football spectator (uncredited)
- El tigre de Chamberí .... Bar customer (uncredited)
- Historias de Madrid .... Bar customer (uncredited)
- La vida por delante .... Nightclub customer (uncredited)
- La violetera .... Man caught up with brawling flower sellers (uncredited)
- Las chicas de la Cruz Roja .... Bar customer / Man at the racetrack (uncredited)
- Malinconico autunno .... Restaurant customer (uncredited)
- Muchachas en vacaciones .... Department store executive (uncredited)
1959 - ¿Dónde vas, Alfonso XII? .... Courtier (uncredited)
- Azafatas con permiso .... Nightclub customer (uncredited)
- Bombas para la paz .... Man at the peace conference (uncredited)
- Échame la culpa .... Man in church (uncredited)
- El día de los enamorados .... Man on train station platform (uncredited)
- El gafe .... Metropolitan bank board member (uncredited)
- Los tramposos .... Executive at the meeting (uncredited)
1960 - 091 Policía al habla .... Man at the airport / Dancing cafe customer (uncredited)
- Ama Rosa .... Party guest (uncredited)
- Días de feria .... Bar customer / Dance contest judge (uncredited)
- El hombre que perdió el tren .... Best man at the wedding / Party guest (uncredited)
- El vagabundo y la estrella .... Party guest (uncredited)
- La reina del Tabarín .... Nightclub customer at New Year (credited)
- La rivolta degli schiavi .... Courtier (uncredited)
- Los económicamente débiles .... Porter (credited)
- Mi último tango .... Journalist (uncredited)
- Pasa la tuna .... Examiner (credited)
- Sólo para hombres .... Man in neighbouring office (credited)
- Suspendido en sinvergüenza .... Employee (uncredited)
- Trío de damas .... Bar customer (uncredited)
- Un ángel tuvo la culpa .... Neighbour / Party guest (uncredited)
- Un rayo de luz .... Parent at the concert (uncredited)
- Y el cuerpo sigue aguantando .... Assistant Doctor (credited as "J. Núñez")
1961 - Armas contra la ley .... Police secretary (credited)
- Don José, Pepe y Pepito .... Lecture audience member (uncredited)
- Ella y los veteranos .... Wedding guest (uncredited)
- Festival in Benidorm .... Journalist (uncredited)
- Fray Escoba .... Man carrying effigy (uncredited)
- La bella Mimí .... (credited, but not found)
- Los pedigüeños .... Man at the funeral (uncredited)
- Margarita se llama mi amor .... School director (uncredited)
- Pecado de amor .... Court notary (uncredited)
- Siempre es domingo .... Doctor (uncredited)
- The Happy Thieves .... Bullfight spectator (uncredited)
1962 - Cupido contrabandista .... Man by the pool (credited)
- Der Teppich des Grauens .... Party guest (uncredited)
- Dulcinea .... Juryman at the inquisition (uncredited)
- El balcón de la Luna .... Nightclub customer (uncredited)
- El grano de mostaza .... Nightclub customer (uncredited)
- El hombre del expreso de Oriente .... Man with the Mayor (uncredited)
- Gritos en la noche .... Nightclub customer (uncredited)
- Historia de una noche .... Man watching the show (uncredited)
- Ipnosi .... Man backstage (uncredited)
- Torrejón City .... Barman (credited)
- Vuelve San Valentín .... Wedding guest (uncredited)
1963 - Chantaje a un torero .... Prisoner (uncredited)
- Cuatro bodas y pico .... Nightclub manager (uncredited)
- El juego de la verdad .... Detective (credited)
- El secreto de Tomy .... Man at the racetrack with Mendoza (uncredited)
- El verdugo .... Man leaving the execution (uncredited)
- La batalla del domingo .... Nightclub customer (uncredited)
- La casta Susana .... Party guest (uncredited)
- La pandilla de los once .... Felipe (credited)
- La Tela de araña .... Man showing the apartment (uncredited)
- La verbena de la Paloma .... Policeman (credited)
- Llegar a más .... (credited, but not found)
- Marisol rumbo a Río .... Waiter on the ship (uncredited)
- Objetivo: las estrellas .... Nightclub customer (uncredited)
- Rififí en la ciudad .... Nightclub customer (uncredited)
- Rocio de La Mancha .... Man in theatre box (uncredited)
- Se necesita chico .... Wedding guest (uncredited)
- Shéhérazade .... Councillor of Haroun-al-Raschid (uncredited)
- Trampa para Catalina .... Member of Paramana embassy staff (uncredited)
- Una chica casi formal .... Hotel desk clerk (uncredited)
1964 - Búsqueme a esa chica .... Theatre audience member (uncredited)
- Como dos gotas de agua .... Dinner guest (uncredited)
- Fuera de la ley .... Donovan (credited)
- Isidro el labrador .... Man at the death bed (uncredited)
- La cesta .... Card player (credited)
- La muerte silba un blues .... Casino patron (uncredited)
- Los dinamiteros .... Mutuality office worker on the stairwell (uncredited)
- Los elegidos .... Neighbour at the window (uncredited)
1965 - Currito de la Cruz .... Priest (uncredited)
- El cálido verano del Sr. Rodríguez .... Neighbour (uncredited)
- El juego de la oca .... Bar customer (uncredited)
- El marqués .... Bar customer (uncredited)
1966 - 7 pistole per i MacGregor .... Arthur (uncredited)
- Hoy como ayer .... Man in 20's flashback (uncredited)
- Frontera Sur (1943) IMDb. This is an Argentine film, so presumably refers to another actor of the same name. I did find a poor quality copy of this one on You Tube and there is an Enrique Núñez credited as playing A Lieutenant. I didn't recognise Enrique, but, with the poor quality, if it is our man and he isn't seen clearly, then it's possible I could have missed him.
- El pendiente (1951) IMDb. This is an Argentine film, so presumably refers to another actor of the same name.
- Los sobrinos del Zorro (1952) IMDb. This is an Argentine film, so presumably refers to another actor of the same name.
- La niña del gato (1953) IMDb. This is an Argentine film, so presumably refers to another actor of the same name.
- A las cinco de la tarde (1960) IMDb as Friend. I didn't find him, but my copy isn't great, so there's a small chance I may have missed him, though it would have been helpful to know whose friend he is supposed to be playing. He isn't credited
- Nada menos que un arkángel (1960) IMDb
- Confidencias de un marido (1963) He appears as a neighbour passing on the stairs, but it would take a good quality copy to get an image where he is identifiable.