The adjective Χριστιανός (‘Christian’) is used three times in the New Testament; twice in Acts and once in 1 Peter.
Here are the phrases in which it is used:
χρηματίσαι τε πρώτως ἐν Ἀντιοχείᾳ τοὺς μαθητὰς Χριστιανούς
called first in Antioch the disciples Christians (Acts 11:26)
Then in Acts 26, Agrippa is speaking to Paul:
Ἐν ὀλίγῳ με πείθεις Χριστιανὸν ποιῆσαι
In little (time) me you persuade a Christian to be?
(“In a short time would you persuade me to be a Christian?”)
And in 1 Peter 4: 16
εἰ δὲ ὡς Χριστιανός, μὴ αἰσχυνέσθω
but if as a Christian (you suffer), do not be ashamed
The word for ‘suffer’ is not present in this phrase, but is assumed from previous context.
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According to the standard lexicons, the term ‘Christian’ was first used by others to refer to the first followers of Christ, not by the followers themselves, but then adopted by them as an accepted title.
There is extensive discussion on the early use of the word ‘Christian’ by the Roman historian Tacitus, a discussion upon which I am in no wise qualified to comment, but if you are interested, the wikipedia article ‘Tacitus on Christ’ seems a reasonable place to start.