Simply put, one player has made the others feel uncomfortable and they no longer want said player to be a part of the campaign. How does one go about asking a player to leave? 

Without knowing further details, if you are the GM, then just let them know about the feelings of the other players. You might gain insight into why they behave the way they do and possibly earn them a second chance.

However if the player is overly disruptive and continues to cause bad vibes, as the GM it is your duty to let them know they are no longer welcome.

If you are a player, then you have to talk to your GM about it.

Unfortunately more details cant be shared but it is not a good situation. The behavior of the player does not warrant a second chance. DM is fully aware of the details and the situation.

Quote from supermomaz >>

Unfortunately more details cant be shared but it is not a good situation. The behavior of the player does not warrant a second chance. DM is fully aware of the details and the situation.

 If that's the case, simply tell them they are not welcome back. You don't want to beat around the bush and leave the message ambiguous with them not knowing if this is just a warning or not. Be frank, firm, and establish that boundary. 

Agreed with prior advice. 

Don't pause, don't wait -- cut it now. 

Be nice, be aware that people don't like to be told they are not welcome, don't react well, will lash out (verbally, usually, but...). 

Quote from supermomaz >>

Unfortunately more details cant be shared but it is not a good situation. The behavior of the player does not warrant a second chance. DM is fully aware of the details and the situation.

 
Since you are not the DM, it is not your place to ask the player to leave - your  DM should be making the decision.  A player might not listen to another player saying “you cannot come back,” but they hardly have a choice if the DM expels them.

You (and the other players, with more people being better) can help the DM in this choice by politely telling the DM that either the offending player leaves or you do.

Politeness is overrated when it comes to people who are feminine hygiene products and the bag in which they came.  Best to be rude and direct, make sure the message is clear and that there is nothing to discuss.  Politeness breeds debate and if the problem is that severe, its best to be super clear about what is going to happen regardless of any further discussion.

Quote from supermomaz >>

Simply put, one player has made the others feel uncomfortable and they no longer want said player to be a part of the campaign. How does one go about asking a player to leave? 

If you aren't the Dungeon Master, then you don't. The DM is in charge of who and isn't in their game, and you can't go behind their back and kick out players because you have no authority or right to do so. Dungeon Masters have the right to find groups that work for them and play the way they want them to, just the same as players do. 

It's very much possible that the DM enjoys what that player is doing, and that you or other players make them uncomfortable too. For instance, in some groups, characters are expected to be criminals, mass murderers, or arsonists. While in others, doing so makes you a problem player.

You've only been able to share less than a paragraph of extremely vague information, so we can't help you too much. But the bottom line at present is that - by definition - the Dungeon Master gets to pick who's in their own gaming group and that doesn't change suddenly when the playing sessions begin. There wouldn't just be a DMing crisis if Dungeon Masters weren't allowed to pick who plays D&D with them; There would literally be next to no DMs because it simply isn't fun to spend hours of work running a game for your friends if you don't have control of whether or not the people you want to play with are allowed to stay in the game this time round.

You do have the right to have a conversation with the Dungeon Master to explain A) What issues you have with the player you want expelled B) Are there any possible compromises here if worst comes to worst and C) What you will do if nothing changes. Ultimately though, it's the DM's choice. And if they already know about this behavior like you say they do, it is very possible that they like or are amenable to it. No player should be able to kick out other ones, and it would suck if the roles were reversed and you were playing Dungeons and Dragons differently than a couple other members of the group, which then got kicked out by those individuals when the person in charge was happy with what you were doing.

You may have to leave, and it's likely the player is doing genuinely awful things or acting in ways that are worse than just playing D&D how they want to, which in itself could be a problem if they didn't talk to the group first about that. Regardless, the advice above - other than what Caerwyn said - seems mildly appalling to me. No wonder so many don't want to run a group if they spend time and effort on getting a campaign going and have their fun shuttered by players kicking other members of the group out.

The DM is responsible for addressing this (unless if you are in a club or a gaming group where members have an equal share in admittance and membership. Then it comes down to a vote and the groups spokesperson must take lead on the decision). 

So the question is what have you informed the DM about this situation and what is your personal course of action based on the DM's response. I am also assuming you have talked to other players, and they feel the same as you? 

As difficult as this situation might be for yourself (and potentially other players) it is more stressful for the DM because they have the responsibility to be the referee of the game, both in and out of session.  So your should be communicating with your DM on this.  Be direct and straightforward; be polite and show respect to the DM (and in your description of the player who is the focus of this issue). 

Now not knowing the situation, so please take that into account when you read this next line:  If the DM opts to not kick the player, then that is their choice and it must be respected.  You have to accept this and make a personal choice from there: find a way to play with this player or leave the campaign and find a new one.  Regardless of how this pans out, just  be honest and direct with expression your concern, accept the decision on how the DM wishes to address this, and then make the best choice for yourself and your gaming experiences, and finally, move forward and leave the past behind.