I am sorry for your loss.
This is straight out of the book, but the final individual income tax return of a decedent is prepared and filed in the same manner as when they were alive. All income up to the date of death must be reported and all credits and deductions to which the decedent is entitled may be claimed.
Publication 559 provides a decent overview of how to handle filing the final individual income tax return for the decedent as well as the income tax return for the estate (if required).
See https://www.irs.gov/publications/p559#en_US_2018_publink100099499
With respect to your father, his final individual tax return (e.g, Form 1040) would be filed under his social security number and he would not pay capital gains tax on the stock he continued to hold through the date of his date (the final return would, however, include stock he had actually sold during the tax year prior to his death).
The basis of the stock he held on the date of his death is stepped up to its fair market value on the date of his death, so that would be the basis of the stock in your hands per Internal Revenue Code Section 1014(a).
Your father's estate would be required to file an income tax return (under the estate's federal tax ID number) if it received gross income for the tax year of $600 or more.
See https://www.irs.gov/instructions/i1041#idm140630132235568
