To some extent, I'm thinking of how this compares to the execution of Charles I by Parliament in 1649, which seems to have caused a far greater stir, yet the process of Richard's deposition from what I know of it through Froissart's Chroniques seems eerily familiar.
Part 1 - The Other Times English Kings were Deposed
The deposition of Richard II was surprisingly uncontroversial at the time precisely because of pre-existing constitutional issues surrounding legitimacy. By the reign of Charles I, there was a clear constitutional arrangement regarding the king's legitimacy, succession etc. that wasn't there in 1399 to cause a stir. It's often said that the removal of Charles I was an unprecedented event, but that really isn't true. The trial and execution was unprecedented, but Charles was not the first English king to be deposed by Parliament. However, Richard's deposition did later lead to legitimacy questions integral to the War of the Roses.
The Kingdom of England had never been great at succession. When Henry I died without a male heir in 1135 it kicked off 18 years of civil war (known as The Anarchy, though it wasn't that bad) between Matilda, Henry's daughter, and Stephen of Blois, his nephew. Almost every medieval English king had some sort of a controversy surrounding succession and legitimacy. Henry II had to deal with repeated attempts to seize the throne of England by his own children. Richard I had problems with John trying to take power whilst on the Third Crusade. The throne of England was notoriously... open to question.
Over time the situation stabilised, thanks in large part to Parliament. Parliament initially existed to solve problems when the king and his advisors felt out of their depth, or were otherwise unable to reach a decision on important matters. During the reign of John and his son Henry, neither of whom could command respect, Parliament became important as a legislative body that did command respect. It gained powers steadily over time, often in vaguely worded treaties that gave it sweeping authority. This meant that Parliament could sometimes step in and exercise power when the king was a bit rubbish. Henry III found this out the hard way in 1258. Henry was widely perceived to be abusing his powers, making poor decisions with the powers he had, and generally leading the kingdom to ruin. On 30 May 1258, a session of Parliament was interrupted as armed men acting on the orders of leading noblemen entered the Palace of Westminster to depose the king. Henry acted fast and entered negotiations immediately - they were already in a session of Parliament so it was a good time to do so. The result of these discussions and further reforms when they met again in June was that Henry would become a ceremonial monarch and his powers would instead be exercised through a council of noblemen appointed by Parliament, known as the Lords Appellant. Henry III would later regain his powers after the Lords Appellant got a bit out of control, but the precedent was set that Parliament had powers above that of the king.
Parliament had come close to deposing the king but through negotiations Henry had just clung to his throne. Edward II (reigned 1307 - 1327) was not so fortunate, though also escaped deposition. Edward II had managed to anger almost every prominent nobleman in England. He exercised power through favoured advisors whom he would shower with gifts. He did not hide his contempt for Parliament, which he saw as just a tax raising institution. Even Edward's own wife, Isabella, openly hated him for a variety of very good reasons and had an affair with the exiled nobleman Roger Mortimer. Isabella and Roger planned to depose the king and raise his son to the throne. Edward II had no support - when he summoned 2000 men to serve in his army only 55 turned up - and his forces surrendered. But he was still king, so Isabella went to Parliament. They were generally rather ambivalent since there was no procedure for actually deposing a king, but protests outside and speeches about the king's failures as a ruler were swaying them into taking action. As they deliberated on what the post-Edward government should look like, more moderate figures in Parliament left the session to speak with the king, urging him to abdicate. They warned that Parliament might decide not to place his son on the throne, so Edward ought to abdicate immediately to ensure that his son (also called Edward) would become king. Seeing that there was no alternative, he did so. The precedent set this time was that Parliament could, if it wanted to, depose the king. It was just that the king in question had pulled the medieval equivalent of 'you can't fire me - I quit!' before they voted to do so.
So when we get to Richard II, another rubbish king, the precedent was there to remove him. Richard had many of the same problems Edward II did. He was authoritarian, cared little for Parliament, and although his decision making was adequate his method of enforcing those decisions was deemed insulting. Like Edward II, he had favourites (which Parliament forced him to remove through the appointment of Lords Appellant, with many executed) and sidelined most of the kingdom. His policy of détente with France was designed to save money and lessen the tax burden on the people, but instead he threw increasingly lavish parties which led him to break his key promise of lower taxation. At least beforehand the money was clearly going on something perceived as worthwhile, but few were happy to pay more tax just so that the king could have an extra boar at dinner. During the 1390s he also seemed to lose his grip on reality to the point where his mental health has been questioned by modern historians, ranging from the generally unfavoured view that he was schizophrenic to Nigel Saul's assessment that he developed a narcissistic personality but did not have a permanent mental affliction. In any case, he started locking up noblemen for no clear reason beyond revenge for previous challenges to his power. Some of these noblemen, like his relative the Duke of Gloucester, mysteriously died in custody. Then, when recruiting for his army in 1398, he started locking up, exiling, or executing people at a local level for no clear reason. The perception was that this was a witch-hunt against people who had criticised him in the past, along with their associates. This was truly insane - he was taking revenge over a power struggle from 10 years ago and ignoring the courts to do so.
In this bloodshed, many opportunistic noblemen sought favour with the king including, oddly enough, Henry Bolingbroke, who was elevated from an earl to the Duke of Hereford. Henry was also one of those Lords Appellant who had curtailed Richard's favouritism a decade earlier, so his position was a difficult one. On the one hand, he was of royal blood (the grandson of Edward III) and a likely successor to the childless Richard II, but his former status as a Lord Appellant also meant he was in line for a sudden imprisonment and mysterious death. Richard didn't move against him in this fashion, probably because killing Henry would have been intolerable. Instead, Henry was exiled and deprived of his lands in 1398. Furthermore, when Henry's father John of Gaunt passed away in early 1399 Richard took John's land for himself rather than letting Henry inherit it. Henry was under no illusions about Richard's plan here - Henry was to be deprived of his status piece by piece until he was nobody. Henry met with the only other surviving Lord Appellant, the former Archbishop of Canterbury, who had also been deprived and exiled. When Richard II was campaigning in Ireland, Henry returned to England to reclaim his lands. Other noblemen sympathised and joined his cause. More still joined him out of fear of his growing army. His support snowballed to the point where people openly discussed the prospect of deposing Richard in favour of Henry. By the time Richard and Henry actually met, Henry had almost all of England loyal to him, and Richard agreed to abdicate the throne to avoid a war. Exactly what happened next isn't clear - some sources say this went smoothly and others claim that Richard actually refused to abdicate once he was imprisoned. Whatever the case, Parliament met on 1st October 1399. 33 articles of deposition were read against the king, they were passed unanimously, and Richard II was deposed. On 13th October, Henry Bolingbroke became Henry IV, establishing the House of Lancaster as the kings of England. Richard then got a taste of his own medicine by mysteriously dying in prison.