[SalesForce] Process Builder Update Parent (lookup) Record Field

You can do something like this. Basically you would need to get the opportunities of all the records that you're inserting / updating, get the associated accounts and then set the account values in the customOject's account__c field.

trigger autoPopulateTrigger on customObject__c (before insert, before update){

    //get the oppotyunity Id's and store it in a set.
    set<Id> opptyIdSet = new set<Id>();
    for(customObject__c co: trigger.new){
        if(co.opportuntiy__c != null){
            opptyIdSet.add(co.opportunity__c);
        }
    }

    //query the opportunity records and get the associated accounts.
    map<id, opportunity> opptyMap = new map<id, opportunity>{[SELECT id, accountid from opportunity where Id IN: opptyIdSet]};

    //update the account value based on the opportunity in the record.
    for(customObject__c co: trigger.new){
        if(opptyMap.containsKey(co.opportunity__c)){
            co.Account__c = opptyMap.get(co.opportunity__c).accountId;
        }       
    }

}

Unfortunately, Process Builder can only fire processes off of changes to the main object. If the Process could potentially be fired off of a change to an Account or a change to a Contact, you'll have to create the Process twice.

For the Contact process, it'll have to fire when the contact change is made and the contact's Account field matches the right criteria. The Account process is trickier. You'll probably have to create a Flow that runs when the Account field is changed, which can look for any contact that matches the Contact field criteria in question and performs the update where it's needed.

Please let me know if you have questions about specifics.