Home routers, such as D-Link ones, generally offer little, if any, monitoring capability, but the question also is what type of monitoring do you want to do? Do you just want to monitor bandwidth utilization or do you want to be able to examine the details of every packet flowing through the router as one could do through a packet analyzer such as WireShark?
Since you didn't mention the model of your router, I don't know what its capabilities might be. I don't know if it even offers any bandwidth utilization statistics or if it offers Simple Network Management Protocol (SNMP) or sylslog capabilities, though some D-Link routers offer those capabilities. E.g., see D-LINK SYSTEMS, INC. | WIRELESS ROUTER : Tools / SysLog. If yours does support syslog, turn it on and set up a syslog server somewhere. It could even be one you set up on your local area network (LAN) at home that you can access from an external location on the Internet. You can find free syslog server software for Linux systems and Kiwi, which is part of SolarWindws offers a free edition of its syslog server for Microsoft Windows sytems. You can view D-Link 704P with Kiwi Syslog Daemon on how to configure one D-Link router for syslog; yours may have a similar setup process. You can see some images of what the Kiwi syslog displays look like here.
Some D-Link routers offer SNMP capability - see How to Configure SNMP on a D-Link. If yours does, you can configure it for SNMP and set up an SNMP manager on a system on your LAN or external to your router on the Internet. For Windows systems, there is a free version of PRTG, which could provide your with statistics and graphs of the bandwidth utilization through the router.
If you want to examine the details of every packet flowing through the router, such as you could do with a packet analyzer, that will be more difficult without replacing the firmware of the router, which you can find explained at How to Use a Custom Firmware on Your Router and Why You Might Want To, though you could, instead, buy a hub and follow the suggestion offered by Tanner Faulkner in response to another Super User question: How can I capture packets going from my router to a specific server?. Hubs have largely been supplanted by network switches, but, doubtless, you could find one on eBay, etc.
Generally, your router has to route traffic to a router belonging to your ISP, which then will route the traffic to another router and the process will continue until a router near the end of the network path is able to deliver the traffic to the destination host. It is unlikely there is any configuration option you can choose in your home router to duplicate all of the traffic passing through it so as to also transmit it to some external IP. Though you might be able to set up some PC at your home location to do so, if you could configure it as an intermediary system between your router router and every system at your home either by making it a transparent proxy or configuring it to be an internal router between your D-Link router and the rest of your home systems. But, in that case, you wouldn't need to send the data to some external system on the Internet; you would just need to give yourself a means to access the system while abroad. E.g., you could use TeamViewer, which is free for personal use, VNC or some other application for remote monitoring on that home system, which sees all of the traffic between your D-Link router and every other home system.
In regards to the link you provided in your comment, I use WallWatcher to monitor an Actiontec router. It works well and you could install it on a Microsoft Windows system either external to your router or on your LAN and forward your router's log to it, if the router supports that option. If the monitoring system is on your LAN, you could use a remote monitoring application to connect to the system running WallWatcher while you are abroad and view information on the traffic flowing through the router. You might be able to see the "from" and "to" IP addresses for traffic flowing through your router by that means. I can see that information via the logs from the Verizon-provided Actiontec router at a site where I use WallWatcher. The key question is whether your router provides that level of detail in its log and whether it can be configured to forward the log to another system.