So, there are several ways to do this: First, translate into English everything except proper nouns (specific names of people, places, or things) or culturally unique concepts, and insulting words or expressive comments (Mein Gott! for a German who is fluent in English) and anything with counting or math that does not need to be done in communications (even if you're a foreign student, most people mentally read numbers and do math in native languages. Even SETI messages are done by making use of numbers to communicate because Math is a more universal language. A common tactic is for the broadcast of a sequence of on prime numbers (Whole numbers that can only be divided by themselves and one 1, 2, 3, 5, 7, 11..) as the concept is very basic in mathmatics, but too random to occur in nature.)
Additionally, slip words of the conlag into dialog in places where the context can give the meaning without explaining it directly. Consider the "Scruffy Haired Nerf Herder" line from Star Wars or the Use of the Klingon word "peta'Q". In both cases, they are not properly defined in their first use, but the times where they are used contextually imply what they are. Starr Wars later confirmed what a "Nerf" is, specifically stating their a disgusting livestock animal, but the audience already knew it was something that was really demeaning to herd because Han responds to the insult by taking offense to the mildly insulting use of "Scruffy" and ignores the implication of being a Nerf Herder. We instantly see this as along the line of someone calling a woman "a cheap whore" to which the woman responds "How dare you call me 'cheap'?". The joke is that they are only insulted when the insult is untrue... if you're gonna be a whore, at least be a high end one, amiright?
Similarly, the word "peta'Q" is frequently used as a fighting word insult in Klingon, with it's frequent use by them and the mild reactions of non-Klingons so insulted to the near murderous rage of Klingons so insulted. It was later confirmed that the word means "dishonored" and the implication is that it's a rather crass way to say it which is why it always comes through the Universal Translator even when the Klingons are speaking English. The best word to describe the contextual meaning of "peta'Q" doesn't translate into anything in English. It would be like a human calling an alien "You Mother f--ker!" and the alien gets the translation of "you father!". The F-Word's practical translation is not carrying the implied meaning of the word.
Prayers and other ritual recitations may also remain intact even if the translation is pretty standard if the religion requires use of a certain language. This is fairly common as Judism, Catholics (until to Vatican II), and Islam all require readings be done in specific language (Hebrew, Latin, and Arabic respectively) and this was common among the religious on Star Trek and one of the few ways the Vulcan and Bajoran languages were used.
Another trick is to not invent new words but a new alphabet and keep the English spelling. You can uses more than 26 characters (I'm fond of paring "Sh", "Ch", "Th", and "Qu" separate symbols than their constituent letters). One thing that commonly comes up that is interesting is that if a cypher-text is spoken, it's commonly held to sound nothing like the words spelled (D.C. comics has two of these languages, in Kryptonian and Interlac. Both use English words and convert each letter to a symbol with a one to one translation, but spoken Kryptonian sounds vaguely Nordic and Interlac sounds equally different.). This does occasionally occur in real langues, for example, Mandarin and Japanese do not sound a like and it's fairly easy to tell the difference upon sound. However, the written Mandarin and Japanese are fairly so similar that a person who can speak and read only one language can still read the other with minimal information loss. Even languages with the same alphabet use different sounds for different letters. For example, English pronounces the letters "U" "V" "W" as "You", "Vee", and "Double-You" respectively, but German pronounces the letters "Oooh", "Vay", and "Double-Vey". Even dialects will change the pronunciation as the letters "H" and "Z" depending on what nation you're from with British English using "Hayche" and "Zed" while American pronunciation being "Ayche" and "Zee" (one British Commedain performing in America drew attention to common British English things Americans get confused by quipped that he pronounces "H" the way he does because "There's a feakin' H in it." He then conceded that the Americans do have a better building floor numbering system where the ground floor is the first floor as opposed to the British, where the Ground Floor was the floor on the ground and the next level up is the first floor, because "That's how Numbers work").
If you're doing a play, one of the best "speaking a different language" gags can be seen in the British Comedy "'Allo 'Allo'" where the actors are all British, but are playing characters centered around French Resistance in a small provincial village during Nazi occupation. All characters speak in a ridiculous accent of their nationality. When necessary for humor, a characters accent will represent the language they are speaking in, so if a French character starts speaking in a German accent English, the other French-accented English speakers will hear him speaking German. Typically these changes will be given a quick acknowledgement by non-speakers as scenes where two different accented characters talk, it's usually assumed both are speaking the same language but the non-native speaker has a notable accent of his/her native language. Typically, the character changing his accent denotes an attempt to sound like a native speaker. Additionally, a running gag was that all English speakers (who use very exagerated English accents peppered with frequent use of "Britishisms" and idioms) were portrayed as fluent French Speakers, but terrible at pronunciations, and depicted this by giving their dialog in French accents, but the dialog was always littered with malapropism (the recurring English Intelligence officer, who was undercover as the towns police officer, would enter every seen with a greeting of "Good Moaning" when he meant to use good morning) and only the French were aware of the poor speaking skills (the Germans always believed he was speaking proper French). The English were always unable to understand any other language while speaking with British accents, so the malapropisms were always their "French Accent" and the show gave different explinations as to why it was consistently mispronounced (the British believed that they were using thick Parisian accents, which the main characters couldn't understand due to their provincial French accents. The French characters just thought it was because the British were idiots. Both are plausible as the British could be using a different dialect, but were still idiots for not realizing most of France is not Paris).
Other works, normally when dubbing into a different language from the work, may depict sterotypes traditionally associated with accents in their native even if characters wouldn't use that accent if they learned the second language fluently. For example, in Japan, most American characters will use an Osakan Accent, even though the Americans who learn Japanese will speak with a Tokyo accent like the Japanese nationals in the work. The reason for this is that the Japanese Sterotype of Americans has a lot in common with that of Osakans (and Japanese media is usually made in Tokyo, so the creative staff will see it as familar). Similarly, when Japanese Media hits the United States, Osakan characters will usually be given Brooklyn or Southern American accents, as their much more hot headed and boisterous characteristics fit with American associations with those accents. Additionally, an overly polite character in Japanese usually gets a Queen's English Accent as the Japanese character is overly using honorifics which to Japanese sounds overly proper. The English language doesn't have honorifics to the extent of the Japanese language, and Americans rarely use the ones they do have. But Americans do see Queen's English accents as being overly proper.
Another example is German accents where the Americans do have an ear for a few different accents, ranging from a Prussian/Norhtern German (Typically these characters are the Nazis who yell "Macht Schnell" or are brutal and efficient and almost mechanical), the southern/Bavarian accent (much more polite and friendly, even when being honest. Typically the Nazi Scientists or the Nazi that insists that they are not barbarians and exude culture. Or Swiss). The other two are more Austrian though rarely called that and are more based on two different individuals speaking styles. The first is "Clearly this is Hitler" and is often loud, grandios, and speaking to orderly mass columns of followers who have already organized into a military formation and with no concept of an indoor voice. The second is "The Arnold" after Arnold Schwarzenegger which range from "Defiantly a killer robot pretending to sound human," to "the one good guy with a German accent" to "clearly Action Movie Hero Arnold in a film that was not written for him to be the lead role" (both Characters were Austrians, though the Austrians are happy to not be associated with the former and the latter is more of an American-Immigrant dream come from a not very remarkable Europe, and become famous, rich, and become the hot guy all the women want to be with and all the guys want to be.). Arnold's accent is also kind of goofy nievity, and it's clearly exaggerated in his later films (Arnold does not talk nearly like that in Real Life these days)... clearly he can act... but he can't get rid of the accent (and it's half the fun of seeing his films).
Interestingly, Germans do see similar accents, but not always in the same light. Where American sees Bavarian Germans as scientists who are more cultured and jovial in than their more barbaric northern/Prussian brothers, Germans see Bavarian accents not to dissimilar to the way English speakers see the American Southern accent (Drunken backwards hicks) and when they dubbed "Hogan's heroes" all German characters in the series (of which only the two regulars were not Nazis) were given ridiculous over the top Bavarian accents... not only pointing out that the Nazi Party formed in the region, but also pointing out that everyone who joined was stupid or drunk or both, and usually very much the first).