In Koine Greek, Matthew 5:48 reads as follows:
ἔσεσθε οὖν ὑμεῖς τέλειοι ὡς ὁ πατὴρ ὑμῶν ὁ οὐράνιος τέλειός ἐστιν.
The key word here is telos, which is often translated as 'perfect' but can equally well be translated "goal, end, or purpose." The telos of a thing is that point to which a thing yearns to be, the purpose for which a thing was built.
It does not imply that perfection has been attained, but it is the trajectory towards which the essence of a thing strives.
In the case of "Be ye perfect," it is completely legitimate to translate telos in the sense
of "Be that which God has designed you to be." As Jude 24 implies, we are being made fit to stand in the presence of God - it is the end for which we are made. Jesus is exhorting us here to be that which God would have eventually be, but he is not necessarily demanding we are already there.
These are the places in which telos is used. The ESV likes to use the word 'mature' to capture this sense.
1 Cor 14:20 NA27 w/GRAMCORD
μὴ παιδία γίνεσθε ταῖς φρεσὶν ἀλλὰ τῇ κακίᾳ νηπιάζετε, ταῖς δὲ φρεσὶν τέλειοι γίνεσθε.
20 Brothers, do not be children in your thinking. Be infants in evil, but in your thinking be mature.
Here, the tie with our behavior is explicit, contrasting the maturity of our walk with the innocence of our understanding in regards to evil.
Phil 3:15 NA27 w/GRAMCORD
Ὅσοι οὖν τέλειοι, τοῦτο φρονῶμεν· καὶ εἴ τι ἑτέρως φρονεῖτε, καὶ τοῦτο ὁ θεὸς ὑμῖν ἀποκαλύψει·
Let those of us who are mature think this way, and if in anything you think otherwise, sGod will reveal that also to you
Here, maturity conveys the sense of growth - the way in which we will grow.
Col 4:12 NA27 w/GRAMCORD
Ἰησοῦ], πάντοτε ἀγωνιζόμενος ὑπὲρ ὑμῶν ἐν ταῖς προσευχαῖς, ἵνα σταθῆτε τέλειοι καὶ πεπληροφορημένοι ἐν παντὶ θελήματι τοῦ θεοῦ. ...that you may stand mature and fully assured in all the will of God
Again, like Jude, the sense of maturity, steadfastness, and being what God designed you to be.
James 1:4 NA27 w/GRAMCORD
ἡ δὲ ὑπομονὴ ἔργον τέλειον ἐχέτω, ἵνα ἦτε τέλειοι καὶ ὁλόκληροι ἐν μηδενὶ λειπόμενοι.
Count it all joy, my brothers, when you meet trials of various kinds, for you know that the testing of your faith produces steadfastness. And let steadfastness have its full effect, that you may be perfect and complete, lacking in nothing.
Here, the fullness of the text probably does the best job of conveying that perfection is a process, begun by suffering, strengthened into steadfastness, and made complete in the perfection - the end state towards which all of this is pointing.