skills

 1. When in disgrace with fortune and men's eyes

Elizabethan coat of arms. Usually a sign of fortune and prosperity.

 1. To be in disgrace (considered to be without honor, be out of favor with)with fortune is presumably to be not favoured by her (taking fortune to be the goddess of 111).
O! for my sake do you with Fortune chide,
The guilty goddess of my harmful deeds,
That did not better for my life provide
Than public means which public manners breeds.
Disgrace is a term which would more usually be applied to a demotion or removal from office. Or to a final humiliation and loss of status. Antony on being defeated by Augustus envisages
The inevitable prosecution of
Disgrace and horror, AC.IV.13.65-6.
In this sonnet the word seems to relate more to a failure to achieve status in the first instance, rather than to a subsequent deprivation.

To be in disgrace (in) men's eyes - this possibly refers to some form of public disapprobation, either real or imaginary. What the disgrace was we cannot say. It could be the mere fact of being associated with the theatre, which by many preachers of the day, and by all Puritans, was considered to be a great den of iniquity and a source of many evils. See the passage at the bottom of this page illustrative of Puritan distrust.

 2. I all alone beweep my outcast state,

 2. beweep = weep for, bewail; Like bewail and beseem, the word has an archaic and biblical flavour.
my outcast state = my condition of being a social outcast. The condition is probably exaggerated for the sake of effect, and to emphahsise that the speaker sees everything in a gloomy light. Fortune has turned against him and he feels that he does not belong any more to society.

 3. And trouble deaf heaven with my bootless cries,
 

 3. deaf heaven - Heaven (God) turns a deaf ear to his complaints and laments. The parallel is drawn with Job in the Old Testament, who was cast out on a dung heap and bewept his mournful state.
bootless = to no avail, achieving nothing.

 4. And look upon myself, and curse my fate,
 

 4. And look upon myself - as the outcast contemplates his own fallen state.
curse my fate - another echo from the Book of Job in the Bible:
After this Job opened his mouth and cursed his day. And Job spake and said: Let the day perish wherein I was born, and the night in which it was said, 'There is a man child conceived'. Let that day be darkness, let not God regard it from above, neither let the light shine upon it. etc. Job.III.1-4.

 5. Wishing me like to one more rich in hope,

 5. Wishing myself to be like one who is more richly endowed with all manner of blessings, including wealth.

 6. Featured like him, like him with friends possessed,

 6. Featured like him, like him = with features like this person, like this second person having friends, like this third, desiring his skills (line 7) etc.

 7. Desiring this man's art, and that man's scope,
 

 7. this man's art = the skill that one particular person has; that man's scope = the capability, range, mental ability that another particular person has.

 8. With what I most enjoy contented least;
 

 8. It is unspecified what he most enjoys, but evidently, in his despondency, things which ought to give him enjoyment do not do so. The implication is that he no longer enjoys the love of his beloved, although that idea is countermanded by the final couplet.

 9. Yet in these thoughts my self almost despising,
 

 9. in these thoughts = while I am engaged in these thoughts
myself almost despising - and almost considering myself to be despicable for being so cast down.

 10. Haply I think on thee, and then my state,

 10.Haply = by chance, by a happy stroke of luck;
my state = my mental state, with a suggestion also that his fortune, or the state of affairs in which he finds himself, improves.

 11. Like to the lark at break of day arising
 

 11. There is an echo of this in Cym.II.iii.20-1
Hark! hark the lark at heaven's gate sings,
And Phoebus 'gins arise...

 12. From sullen earth, sings hymns at heaven's gate;

 12. sullen = gloomy, dark, miserable;
From sullen earth - the phrase may be taken both with this and with the preceding line. The lark rises from sullen earth, and it also sings hymns which rise up from the earth to the gate of heaven, or, as it sings, it rises from earth towards heaven.
sings - the subject is the lark, but also the poet's soul, which has been liberated by his thinking of his beloved.

 13. For thy sweet love remembered such wealth brings

 13. thy sweet love remembered = when I have called to mind your love, when your sweet love springs up again in my memory.

 14. That then I scorn to change my state with kings.

 14. Although the primary meaning is that 'I am happier than a king could be, and therefore have no wish to swap places with him' there is a hint of the political meaning of state, i.e. nation state, as in 64:
When I have seen such interchange of state,
Or state itself confounded to decay;
Hence, 'even though I were to have a kingdom, I would not exchange it for the the happiness of knowing you'