Depends on the period but assassins and paid killers never really fall out of style. However, one could always take the matter into your own hands (judiciously or extra-judiciously).

John the Fearless, duke of Burgundy, wanted to remove a political rival, Louis of Orleans. John hired a group of assassins and placed them in a house near the royal hôtel de St. Pol. The assassins were led by a Norman knight called Raoul d’Anquetonville, who apparently had a fair amount of gambling debt. A French clerk called Nicolas de Baye describes it thus:

Louis, son of King Charles V and brother of King Charles now ruling, duke of Orleans (...) accompanied, modestly enough for someone of his rank, by three mounted men and two on foot (...) was struck down and killed by eight or nine armed men who had hidden in a [neighbouring] house

This example comes from both the late end of the medieval period and the highest end of the nobility. John the Fearless would be himself murdered on the bridge of Montereau on 10 Sept. 1419 by the Armagnac faction which Louis had once led. The King's son, the Dauphin Charles, was also present but his level of involvement is debatable.

The problem was often not that you couldn't kill someone, which could be fairly easily achieved, but that someone would come after you. Feud was a major issue across many societies and periods, from Saga Iceland to 1990s 'gangland' America.

With the right planning it is no more difficult to kill a person today than it was a thousand years ago. The methods have changed slightly, but you can still have someone cut down in the street or hire a hit man to walk up to their door and shoot them in the face. In certain places the nobility had an inherent right to feud between themselves (although not within a feudo-vassalic relationship). France was an example of such a society, the thirteenth-century jurist Philippe de Beaumanoir explained this 'right' thus:

if the gentlemen made war on the townsman or commoners, and townsmen or commoners could not make war on gentlemen, they would soon be dead or in terrible trouble (mal bailli).

Recourse to law was aimed at ameliorating the most damaging elements of a blood-feud. In medieval Wales the cenedl (kindred) acted collectively when paying or receiving a galanas (an emendable fine for homicide). Wales was notorious for feuds and when travelling, according to Gerald of Wales, it was common not to ask for names lest you realise you were currently feuding. The galanas was a penalty upon the whole kindred should one of their own commit homicide, and acted as recompense for the aggrieved to prevent retaliation.

Of course the best way to avoid a feud, fine, or trial is to hire an outsider. So while they were probably not professional assassins, there were plenty of trained soldiers who'd be willing to kill for pay.

• The Treatise on the Laws and Customs of the Realm of England Commonly Called Glanvill, ed. and trans. G.D.G. Hall, Claredon Press, Oxford, 1993.

• Philippe de Beaumanoir, Coutumes de Beauvaisis, trans. F.R.P. Akehurst, University of Pennsylvania Press: Philadelphia, 1992.

• Paul Hyams, Rancor and Reconciliation in Medieval England, Cornell University Press, London, 2003.

• Richard Kaeuper, ​Chivalry and Violence in Medieval Europe, Oxford University Press, Oxford, 1999.

• Richard Vaughn, John the Fearless: the growth of Burgundian power, Longman, London, 1979.