Overview
Who, while being in the form of God, did not regard the being equal with God a thing-to-be-grasped (DLNT)
ὃς ἐν μορφῇ θεοῦ ὑπάρχων οὐχ ἁρπαγμὸν ἡγήσατο τὸ εἶναι ἴσα θεῷ
The Disciples Literal New Testament uses form which is probably the better word-for-word treatment of μορφῇ. Note however, the DLNT italicizes the and of to show the translator understood them as ellipses. The literal Greek is morphē theou: there is no article. In other words, the literal Greek differs from an English interpretation which understands the form/appearance/nature of the God. Also, theos is in the genitive, and theou can mean of God or from God.
If morphē is taken as form or appearance, that hardly settles the meaning, because how does the form or appearance of God exist apart from the nature of God? In other words, it is true morphē most likely envisions that which is external, but how can one have morphē theou, which is external of God, independent of that which is internal, or by nature God? Arguably, the only way one has the external form or appearance of God is by also possessing the nature of God. Conceptually, neither form, appearance, or nature do justice to the phrase Paul has constructed. Any attempt to use the lexicon to understand the intended meaning as either form, or appearance, or nature will fall short of conveying Paul's meaning.
Glory
What form or appearance would agree with Paul's terminology? Contextually, the outward appearance would be glory which is how the passage begins and ends:
Let nothing be done through strife or vainglory; but in lowliness of mind let each esteem other better than themselves. (2:3 KJV)
μηδὲν κατ᾽ ἐριθείαν μηδὲ κατὰ κενοδοξίαν ἀλλὰ τῇ ταπεινοφροσύνῃ ἀλλήλους ἡγούμενοι ὑπερέχοντας ἑαυτῶν
and every tongue should confess-out that Jesus Christ is Lord, to the glory of God the Father. (2:11 DLNT)
καὶ πᾶσα γλῶσσα ἐξομολογήσηται ὅτι κύριος Ἰησοῦς Χριστὸς εἰς δόξαν θεοῦ πατρός
Paul begins by encouraging the Philippians to do nothing out of self-seeking glory and ends with the phrase δόξαν θεοῦ πατρός, glory theou Father. Understanding morphē theou as glory agrees with what Jesus said of Himself:
4 I glorified you on earth, having accomplished the work that you gave me to do. 5 And now, Father, glorify me in your own presence with the glory that I had with you before the world existed. (John 17 ESV)
Philippians 2 fits what is found in the Fourth Gospel. Jesus had morphē theou, glory with the Father before the world was. He emptied Himself of this glory in order to take on the form of a slave. In this case, Paul is explaining how Jesus' request was answered.
9 Therefore God also highly-exalted Him, and granted Him the name above every name 10 in-order-that every knee should bow at the name of Jesus— of heavenly ones and earthly ones and ones under-the-earth— 11 and every tongue should confess-out that Jesus Christ is Lord, to the glory of God the Father. (DLNT)
In addition to receiving back that which He emptied Himself of, He was more highly exalted.
Holiness
What "nature" of God does Jesus have? He was without sin (cf. Hebrews 4:15). Being without sin, He would be holy. The sense of being holy is to be separate. Since sin separates man from God who is holy, Jesus emptied Himself of that which is by nature separates man from God. This was accomplished by taking on human form, which is the from of a slave (cf. Romans 6:16-19). That is, He who was without sin and was holy had the appearance of a sinner who by nature had the appearance of being separated from God.
Conclusion
Based upon the New Testament, form, appearance, and nature all fit Paul's use and can reasonably used to translate the phrase. If there is any inaccuracy in translating morphē theou as "nature of God" it lies in capturing only part of the full meaning. Yet the same could be said of "form/appearance of God." If one was to use a single word, in my opinion it would be "glory."