Well you have lots of SOQL statements inside for loop, whihc is not a good practice. You may consider using collections to avoid soql inside for loops. I am highlighting the problem areas in your code.

public class PopulateCorporateFields {

    public List < Corporate_Account_List_Assignment__c > calist {
        set;
        get;
    }

    public PopulateCorporateFields(ApexPages.StandardController controller) {

    }

    public void updateFields() {
        calist = [SELECT Account_Name__c, Corporate_Partnership_Director_1__c, Corporate_Partnership_Director_2__c, Corporate_Partnership_Director_3__c, Corporate_Partnership_Director_4__c, Corporate_Partnership_Director_5__c, Marketing_Solutions_1__c, Marketing_Solutions_2__c, Marketing_Solutions_3__c, Media_Services_1__c, Media_Services_2__c, Media_Services_3__c, Digital_Manager_1__c, Digital_Manager_2__c, Digital_Manager_3__c, Digital_Manager_4__c, Digital_Manager_5__c, Id, Name, Ultimate_Parent_Name__c FROM Corporate_Account_List_Assignment__c];

// IMPORTANT : The above query doesnt have any filters/where clause and hence will bring in all the data//


        for (Corporate_Account_List_Assignment__c ca: calist) {
            ca.Corporate_Partnership_Director_1__c = null;
            ca.Corporate_Partnership_Director_2__c = null;
            ca.Corporate_Partnership_Director_3__c = null;
            ca.Corporate_Partnership_Director_4__c = null;
            ca.Corporate_Partnership_Director_5__c = null;

            ca.Digital_Manager_1__c = null;
            ca.Digital_Manager_2__c = null;
            ca.Digital_Manager_3__c = null;
            ca.Digital_Manager_4__c = null;
            ca.Digital_Manager_5__c = null;

            ca.Marketing_Solutions_1__c = null;
            ca.Marketing_Solutions_2__c = null;
            ca.Marketing_Solutions_3__c = null;

            ca.Media_Services_1__c = null;
            ca.Media_Services_2__c = null;
            ca.Media_Services_3__c = null;
            System.debug('******ID********' + ca.Id);

//ALL these query are inside for loop, and hence adding upto the limit
            Account ac = [SELECT Name, ParentId, Id FROM Account where Id = : ca.Account_Name__c];

            List < AccountTeamMember > atmcpd = [SELECT Id, TeamMemberRole, UserId, IsDeleted FROM AccountTeamMember WHERE AccountId = : ac.Id AND TeamMemberRole = 'Corporate Partnership Director'];

            List < AccountTeamMember > atmMS = [SELECT Id, TeamMemberRole, UserId, IsDeleted FROM AccountTeamMember WHERE AccountId = : ac.Id AND TeamMemberRole = 'IMS'
                AND Isdeleted = FALSE
            ];

            List < AccountTeamMember > atmMDS = [SELECT Id, TeamMemberRole, UserId, IsDeleted FROM AccountTeamMember WHERE AccountId = : ac.Id AND TeamMemberRole = 'Media Services'
                AND Isdeleted = FALSE
            ];

            List < AccountTeamMember > atmDM = [SELECT Id, TeamMemberRole, UserId, IsDeleted FROM AccountTeamMember WHERE AccountId = : ac.Id AND TeamMemberRole = 'Digital Manager'
                AND Isdeleted = FALSE
            ];
            if (atmcpd != null || atmcpd.size() > 0) {
                // User u = [select Id,Name from user where Id=: atm.get(0).UserId];
                //System.debug('******Size*******'+atmcpd.size()+'**********'+atmcpd.get(0).UserId);
                if (atmcpd.size() > 0)
                    ca.Corporate_Partnership_Director_1__c = atmcpd.get(0).UserId;

                if (atmcpd.size() > 1)
                    ca.Corporate_Partnership_Director_2__c = atmcpd.get(1).UserId;

                if (atmcpd.size() > 2)
                    ca.Corporate_Partnership_Director_3__c = atmcpd.get(2).UserId;

                if (atmcpd.size() > 3)
                    ca.Corporate_Partnership_Director_4__c = atmcpd.get(3).UserId;

                if (atmcpd.size() > 4)
                    ca.Corporate_Partnership_Director_5__c = atmcpd.get(4).UserId;
            }

            if (atmms != null || atmms.size() > 0) {
                if (atmms.size() > 0)
                    ca.Marketing_Solutions_1__c = atmms.get(0).UserId;

                if (atmms.size() > 1)
                    ca.Marketing_Solutions_2__c = atmms.get(1).UserId;

                if (atmms.size() > 2)
                    ca.Marketing_Solutions_3__c = atmms.get(2).UserId;

            }


            if (atmmds != null || atmmds.size() > 0) {
                if (atmmds.size() > 0)
                    ca.Media_Services_1__c = atmmds.get(0).UserId;

                if (atmmds.size() > 1)
                    ca.Media_Services_2__c = atmmds.get(1).UserId;

                if (atmmds.size() > 2)
                    ca.Media_Services_3__c = atmmds.get(2).UserId;

            }
            if (atmdm != null || atmdm.size() > 0) {
                if (atmdm.size() > 0)
                    ca.Digital_Manager_1__c = atmdm.get(0).UserId;

                if (atmdm.size() > 1)
                    ca.Digital_Manager_2__c = atmdm.get(1).UserId;

                if (atmdm.size() > 2)
                    ca.Digital_Manager_3__c = atmdm.get(2).UserId;

                if (atmdm.size() > 3)
                    ca.Digital_Manager_4__c = atmdm.get(3).UserId;

                if (atmdm.size() > 4)
                    ca.Digital_Manager_5__c = atmdm.get(4).UserId;
            }
        }
        update calist;
    }

}

 Have a look here on how to bulkify your code to avoid too many soql : http://wiki.developerforce.com/page/Best_Practice:_Bulkify_Your_Code