The answer to this seemingly simple question is actually quite complicated. It begins with Queen Elizabeth I and the Queen's Men and ends with King James I and the King's Men. The simple and most popular answer is: The troupe of actors that Shakespeare is associated with, as both an actor and a playwright, is the Lord Chamberlain's Men.
The Full Story
The full story involves acting troupes that have name changes when (1) the noble or royal patrons of the troupes undergo changes to their titles or when (2) troupes reorganize under different noble or royal patronage [patronage: support and financial gifts from wealthy patrons to participants in the arts]. It starts, as I said, with Queen Elizabeth I.
Queen Elizabeth I organized the first licensed acting troupe in 1583; it was licensed under her new licensing law and was called the Queen's Men, since all actors then were men.
1. Pembroke's Men: Before 1592, which is the year theatres in London closed because of an outbreak of plague (a contagious disease that has to be contained by restricting gatherings, just as the US restricted gatherings in 2009 when there was an outbreak of H1N1 swine flu), Shakespeare is said to have been both an actor and a playwright for Pembroke's Men, sponsored by their patron Lord Pembroke, Henry Herbert, 2nd Earl of Pembroke, and organized under his patronage in 1591-92. The troupe was disbanned in 1592 because of the plague.
2. Lord Strange's Men / Derby's Men: It is probable that Shakespeare joined Lord Strange's Men, of Ferdinando Stanley, Lord Strange (pronounced strang), and performed with them at Rose Theatre (not closed by edict of the Queen) until 1593. Strange's Men were the first to perform Shakespeare's Richard III and Henry VI Part 2. It is also thought probable that Shakespeare used this period of time to focus on his poetry, publishing his first poetic work, Venus and Adonis, in 1593. Also in 1593, Lord Strange inherited his father's title and became 5th Earl of Derby, whereupon his acting troupe came to be called Derby's Men.
3. Lord Chamberlain's Men: Derby had hoped to become regional Chamberlain of Chester but, when Thomas Egerton was instead made Chamberlain of Chester in 1593, the year before theatres in London were reopened, Derby's acting troupe, Derby's Men, was reorganized, meaning some actors went to a different acting troupe, specifically, to Lord Hunsdon's--Lord Chamberlain of the Queen's Court--acting troupe (the Court of Elizabeth appointed more than one Chamberlain, with appointees at the regional level, e.g., Chamberlain of Chester, and at the London Court level, i.e. Lord Chamberlain (Hunsdon)).
Lord Hunsdon's acting troupe, the Lord Chamberlain's Men, was reorganized when Shakespeare moved his allegiance from the disappointed Earl of Derby's (Lord Strange's) Derby's Men to Lord Hunsdon's Lord Chamberlain’s Men, sponsored by their patron Henry Carey, 1st Lord Hunsdon and Lord Chamberlain.
4. Lord Hunsdon's Men: Upon the death of Henry Carey Lord Hunsdon in 1596, patronage of the Lord Chamberlain's Men moved to his son, George Carey, 2nd Lord Hunsdon (he did not inherit his father's position as Lord Chamberlain), and the players were called by their previous title of 1585, Lord Hunsdon's Men.
5. Lord Chamberlain's Men: Another twist places Shakespeare back with the Lord Chamberlain's Men when George, 2nd Lord Hunsdon, was himself appointed by Elizabeth I to the post of Lord Chamberlain in 1597, at which time Hunsdon's Men were accordingly renamed (again) the Lord Chamberlain's Men.
6. The King's Men: When King James I of England, and VI of Scotland, ascended to the English throne in 1603, by the issuance of royal letters patent (a legal document issued by a monarch, similar to a proclamation issued by a president), he assumed the patronage of the Lord Chamberlain's Men who then became the King's Men.
The group was originally called The Lord Chamberlain's Men but the group went through two name changes and were additionally named for Hunsdon and James I. The name was first changed to Lord Hunsdon's Men the to. According to Wikipedia:
The Lord Chamberlain's Men was the playing company that William Shakespeare worked for as actor and playwright for most of his career. ... it had become, by 1603, one of the two leading companies of the city and was subsequently patronized by James I.
... it was briefly known as Lord Hunsdon's Men for George Carey, 2nd Baron Hunsdon, until Hudson in turn became Lord Chamberlain ... whereupon it reverted to its previous name. The company became The King's Men ... when King James ascended the throne ....