public class LeadTriggerHandler {

public void BeforeInsert(List<Lead> newList){
    List<Lead> leadsWithFive9InboundCampaign = new List<Lead>();
    for(Lead l : newList){
        if(l.Five9_Inbound_Campaign_Name__c != null){
            leadsWithFive9InboundCampaign.add(l);
        }
    }
    if(!leadsWithFive9InboundCampaign .isEmpty()){
        populateLeadSourceNameLookup(leadsWithFive9InboundCampaign );
    }
}

public void BeforeUpdate(List<Lead> newList, Map<ID,Lead> newMap, List<Lead> oldList, Map<ID,Lead> oldMap){
    List<Lead> leadsWithChangedFive9InboundCampaign = new List<Lead>();

    if(!leadsWithChangedFive9InboundCampaign .isEmpty()){
        populateLeadSourceNameLookup(leadsWithChangedFive9InboundCampaign );
    }
}

private void populateLeadSourceNameLookup(List<Lead> myLeads){
    Set<String> five9CampaignNames = new Set<String>();
    for(Lead l : myLeads){
        if(l.Five9_Inbound_Campaign_Name__c == null){
            l.Lead_Source_Name__c = null;
        } else {
            five9CampaignNames.add(l.Five9_Inbound_Campaign_Name__c);
        }
    }
    Map<String,Id> campaignNameToId = new Map<String,Id>();
    for( Campaign c : [SELECT ID, Name FROM Campaign WHERE Name IN :five9CampaignNames ]){
        campaignNameToId.put(c.name,c.id);
    }
    for(Lead l : myLeads){
        if(l.Five9_Inbound_Campaign_Name__c != null){
            l.Lead_Source_Name__c = campaignNameToId.get(l.Five9_Inbound_Campaign_Name__c);
        }
    }

}

Method does not exist or incorrect signature: void AfterInsert(List<Lead>, Map<Id,Lead>) from the type LeadTriggerHandler

Method does not exist or incorrect signature: void AfterUpdate(List<Lead>, Map<Id,Lead>, List<Lead>, Map<Id,Lead>) from the type LeadTriggerHandler

Method does not exist or incorrect signature: void BeforeDelete(List<Lead>, Map<Id,Lead>) from the type LeadTriggerHandler
Method does not exist or incorrect signature: void AfterDelete(List<Lead>, Map<Id,Lead>) from the type LeadTriggerHandler

trigger LeadTrigger on Lead (before insert, before update, before delete, after insert, after update, after delete) {

LeadTriggerHandler handler = new LeadTriggerHandler();


if(Trigger.isInsert && Trigger.isBefore){
    handler.BeforeInsert(Trigger.new);
}
if(Trigger.isInsert && Trigger.isAfter){
    handler.AfterInsert(Trigger.new, Trigger.newMap);
}
if(Trigger.isUpdate && Trigger.isBefore){
    handler.BeforeUpdate(Trigger.new, Trigger.newMap, Trigger.old, Trigger.oldMap);
}
if(Trigger.isUpdate && Trigger.isAfter){
    handler.AfterUpdate(Trigger.new, Trigger.newMap, Trigger.old, Trigger.oldMap);
}
if(Trigger.isDelete && Trigger.isBefore){
    handler.BeforeDelete(Trigger.old, Trigger.oldMap);
}
if(Trigger.isDelete && Trigger.isAfter){
    handler.AfterDelete(Trigger.old, Trigger.oldMap);
}