This is adapted from an older answer of mine, but I'd be happy to field follow ups as well!
So the stage is set, the body is in the street, bloodied dagger some yards away, and you, the guilty party, have fled across town (probably not all that far truth be told given the size of London at this time [though it is still by far the largest city in England]).
Now the time frame that you've specified places us in Norman England though early enough that the majority of the populace was still using Anglo-Saxon law codes and practices; we don't really get into major Norman overhauls of the legal code until the 12th century. London is a little bit of a weird case, but we can be general enough to lay out some possibilities.
For the sake of simplicity I'm going to assume that you and your victim are both free (slavery was still around legally for at least 5 more years in England) able bodied men engaged in some local craft. So you've killed someone, left the body in the streets, and made it back to your dwelling seemingly undetected, what happens next? Well obviously someone will find the body in the morning and report it to a local authority. Who would that be?
Now for much of the world until modern times there was no independent (ostensibly) and investigative group of institutions that sought to enact, enforce, and execute the law. Modern media translates modern municipal police departments into medieval-esque institutions such as the "town guard" (think the Skyrim Hold guards), but is that how it worked in the actual Middle Ages? In short, no. Nor would soldiers of any description be permanently stationed within the city of London at this time for the express purpose of hunting down and investigating criminals. So the people who are going to be looking for you are not the medieval equivalent of detectives and beat cops. Instead word will likely spread until someone directly connected with the victim can identify the body. Though people might simply know them based on sight depending on how well known the person was in the community.
In Anglo-Saxon England vigilantism, not a town guard or much less police force, was more or less the "default" method of conflict resolution between equal members of society. There was no real distinction between vigilante justice and "official" justice, nor did independent courts, lawyers, etc... exist at this point in England's history. Especially in the Early Anglo-Saxon period our ideas of objective justice and an independent judiciary/police force just simply were not a part of society. Justice and revenge were extremely interlinked, and the line between official prosecution and a lord taking justice "into his own hands" was significantly blurrier than we might imagine.
So the kinds of people who would be investigating the murder would be the people who were responsible for/to the dead person in some way. These could be their surviving relatives such as brothers, sons, fathers, in-laws, or it could be someone connected to them by social/political bonds, if you've killed a thane or some other dependent of a local big wig he might get involved and round up some others to look around. Let's say that your hapless victim's family is made aware of their death and have rounded up some cousins, people who owe them favors, and others to look around and find out what happened, your victim's local patron has also been notified. So these people who were tied to the victim in some way would then start essentially an investigation, this could be questioning potential witnesses, tracking down people known to have had problems with the deceased and so on.
But how could these people find you specifically? You've gotten away scott free, or so you think. While you may have discarded the weapon and left the scene of the crime unseen, did you arrive home unseen? You probably don't live alone in a medieval city so who saw you come into your dwelling at some ungodly hour? Did they see the blood on your clothes that you surely have from the murder (if you are not extremely wealthy you probably don't have tons of spare clothes lying around to change into)? Do you have nosy or chatty neighbors who might rat you out? And let us not forget the murder weapon!
You like any other self respecting Anglo-Saxon well to do man probably have a knife on you most of the time, and its probably what you used in the murder. So know you're out a knife that people might recognize as yours that is alongside the dead body, and your neighbors might be a little suspicious if you were making a noise at some freakishly early hour. So things aren't looking good, but this is all circumstantial right? Well...that might not matter much
All of this would add up and make you a figure of suspicion, but in the absence of damning obvious evidence what would happen next? Unfortunately for you, your personal weapon has been found at the scene of the crime and one of your neighbors recalls that you came home very late and in something of a rush with blood all over your clothes. So things aren't looking good.
Your victim's brother formally brings a suit against you at the next meeting of the local hundred meeting (basically a monthly community court/assembly) and accuses you of murder, what happens next?
Well this depends a great deal on a variety of factors. Do you want to admit that you killed the man, but there were extenuating circumstances? Did he owe you a great debt? Had he recently insulted you? Was he involved in a crime against you or your family? All of these factors would be taken into account and could change both the way the "trial" is handled and potential consequences you might face. In that case local notables such as important peers such as other craftsmen, the local churchmen, and a representative of the king would likely convene and decide on a sentence that befits the crime and any extenuating circumstances. This could be exile, the death penalty, or if the extenuating circumstances were strong enough, you could instead be ordered to undertake penance instead.
But lets say that you deny everything, you've never shared more than a word with the man, your knife was stolen a week ago, and you were arriving late because you were out and about on business. So its your word against the word of the people accusing you.
Now this is where it can get complicated and it really matters who you know and how much they like you. You have the option of essentially calling in a series of character witnesses to swear that you would never be involved in such a heinous crime. These kinds of people would be your relatives, in-laws, your patron and his networks, etc... If you are a man of good repute with no prior black marks against your honor, and able to call in some favors from your own influential patrons, this will probably be the end of the matter...unless it can be demonstrated that you're lying.
This would be a huge problem for you as you have now sullied not only your own honor by lying, but the honor of those who stood up for you. This could entail its own punishment (likely some form of mutilation of your face or mouth alongside not being able to call witnesses or bear witness in court in the future)
However if you cannot call in these favors, or if you have previously run afoul of the law before, the number of witnesses you need will be higher and you might be in trouble. You can appeal to an ordeal instead if you'd like to try and play for time or put the matter in the hands of God, but the Church and local authorities might not allow this if they think you are plainly guilty and trying to get out of punishment. The ordeal is a process by which you must plunge your hand or (if you have an ill reputation) your arm into a cauldron of boiling water and retrieve a stone. The would would be examined some time later to determine if God had said you were guilty or not. Now this is actually probably not what would actually occur however. If you invoked an ordeal, it was more like declaring an appeal to the Church according to Peter Brown, and in the time before the actual ordeal itself the Church would try and find some agreement between the two parties.
This could be something like a monetary payment, dependent on the social status of the person you killed, and an agreement to undergo penance as well as swearing an oath to not pursue the matter further or try to take further violent action.
So if you decide to not go with an ordeal, the assembly will likely render a judgement based on the evidence presented, the witnesses (both in the sense of actual witnesses and character judges/patrons), and taking your own history into account. The end result, especially in Norman times, is more likely to be physical punishment up to and including death, rather than the earlier practices of paying wergeld (literally man price, or the amount of money that was attached to people depending on their station in life). Now depending on if you've actually been physically restrained or apprehended, you might try and escape, and this would have you declared an outlaw, able to be killed on sight by anyone, little more than a wolf as far as the law was concerned.
Now notice what isn't mentioned here, evidence, trials by lawyers, juries of your peers, judges evaluating evidence, precedent, and case law. All of these hallmarks of modern law quite simply did not exist. Indeed, the actual fact of your guilt isn't really as important so much as your ability, or the suing party's ability, to claim innocence and have it backed up by other members of good standing, the actual facts, evidence (rudimentary as methods the of collecting it were), and so on. What matters is not your actual guilt, but your ability to be convicted based on your social reputation and that of your victim.