Starting off, the author self-identifies as Paul. While there is debate on whether the Paul wrote this letter, that's outside the realm of this discussion. So for shorthand, I will call him 'Paul', regardless of who the author actually was.
Literally, the word παραγγελιας refers to a 'command', 'mandate' or 'instruction'. It is an authoritative order being given. This is the sense of the word (and its root verb παρηγγελω) in other passages of the Septuagint1 and the New Testament.2 The Perseus Digital Library shows it's a relatively rare word, but likewise summarizes it as a 'command'. Both LSJ and Middle Liddell give a primary definition as 'a command or order issued to soldiers'. This will come in handy later on.
Fortunately, the word (and its relatives) is used 34 times in the New Testament books, seven of those instances being found in 1 Timothy: 1.3, 1.5, 1.18, 4.11, 5.7, 6.13, and 6.17. All of these are in the context of instructing or commanding someone else. Since the noun and verb are together used three times in 1 Timothy 1, we clearly need to read the whole chapter to understand what the 'charge' (παραγγελιας) of 1.5 actually is.
In verse 1.3, Paul explains that he left Timothy in Ephesus to 'charge' (παραγγειλης) people of that church. In verse 1.18, Paul states that he has entrusted 'this charge' (παραγγελιαν) to Timothy. Thus 1.3 and 1.18 form an inclusio; the 'charge' being discussed in 1.5 is actually discussed from 1.3 all the way to 1.18.
Verse 1.3 has him say the 'charge' is opposed to 'different teaching'. He initially explains that this 'different teaching' (ετεροδιδασκαλειν) consists of 'stories and endless genealogies'.3 However, in verse 1.10, he goes on to list various types of sins, saying they are opposed to 'healthy teaching'. (In other words, the listed sins in 1.9-10 are the 'different teaching' from 1.3.)
In contrast to all this, Paul says in 1.5 our 'charge' (παραγγελιας) is love, 'good conscience' (συνειδησεως αγαθης), and 'genuine faith' (πιστεως ανυποκριτο). After a high doxology in 1.17, Paul repeats the sentiments of 1.5 over in 1.18-19, where he says 'this charge I deposit to you', so that Timothy may 'serve the good war' (here is the sense of a 'charge' being a military command, as mentioned above), with 'faith' (πιστιν) and 'good conscience' (αγαθην συνειδησιν).
However, chapter 6 has several significant parallels to chapter 1, mostly in the same order.
In verse 6.3, Paul again uses the word 'different teaching' (ετεροδιδασκαλει), repeating verse 1.3. (Verses 6.20-21 contain a reiteration of the 'different teaching' described in 1.6-7.) This 'different teaching' is contrary to Jesus' own words, and only causes arrogance and arguing (6.3-5). This brings us to the major difference between chapter 1 and 6. Chapter 1 includes that list of various types of sins, which are the 'different teaching'. In its place, chapter 6.11-14 instead lists a series of virtues, which Paul 'charges' (παραγγελλω) Timothy to keep. The parallels with chapter one strike back up again, with Paul providing a second high doxology in 6.15-16 (as 1.17), and again telling Timothy to 'guard the deposit' in verse 6.20 (as 1.18).
Going from the inclusio in 1.3-18, and the later parallels in 6.3-21, the 'charge' includes love, good conscience, genuine faith (1.5b,19), and righteousness, reverence, faithfulness, love, endurance, gentleness (6.11). In other words, the context implies the 'charge' is the Christian faith itself, exemplified by how Christians are to act in their day-to-day life.
This seems clear when we consider what the opposite of the 'charge' is, the 'different teaching' that is mentioned in chapters 1 and 6. The type of behavior Paul saw as intrinsically opposed to the 'charge' included: lawlessness, insubordination, irreverence, sin, unholiness, profaneness, patricide, matricide, murder, immoral sexuality, enslaving, lying, oathbreaking (1.9-10), and jealousy, arguing, blasphemy, speculation, evil (6.4), among other things.
1 e.g. Joshua 6.7, 'Instruct the people to go around'; 1 Samuel 22.14, 'and [who is] chief over all your commands?'
2 e.g. Matthew 10.5, 'These twelve Jesus sent, charging them'; Luke 5.14, 'He commanded him to tell no one'; 1 Corinthians 7.10, 'To the married I give this instruction'.
3 I take Paul here to be arguing against using mythologized propaganda to defend the Christian faith, as was common practice in various cultures at the time. e.g. Julius Caesar claimed to be a descendant of the goddess Venus, through the hero Aeneas, who fought in the Trojan War and allegedly helped to establish Rome. Thus, Caesar's ancestry was not only a vital part of Rome's very existence (Aeneas), but his ancestry was divine (Venus). This ancestry qualified him to rule Rome.