Easiest and safest way to get from Gare du Nord train station to Rue de Berri

Being a tourist
Nothing wrong with looking like a lost tourist at a station if you are a lost tourist at a station. Most people will understand and help you find where you need to go, if only by saying "Taxi" and pointing the right way. Just do not accept iffy offers to transport you in a private car, people just do not do that in main cities like Paris.
As you can see in the answer by @abligh, people do offer rides for money, sometimes much more money than the taxi would cost you. And more nasty things do happen by illegal 'taxi' drivers.

Taxis
In a main city like Paris, taxis are around all the time and easy to use.
If you find the queue for the taxis is too long you can take alternative transportation, like buses and the metro, but you will need to work out where to go, which bus or metro to take and where to catch it.
Within the train station there are signs to the taxi ranks, as far as I remember.

Uber and alike
Uber has been available in Paris on and off for a while, but with a lot of controversy and conflict (at times violent) with both taxi drivers and city authorities. Googling uber in Paris brings up plenty of discussion of the situation; it sounds like it’s running and comparatively calm at the moment.
(Thanks to @PLL who posted this as a comment.)
If you are an Uber user, or use one of the other companies that might run in Paris, use your app to see whether something is available.

Public transport map
If you want to use public tranport, here is a link to an interactive map.
Use the 'instructions for use' (tab in the top of the page) as the map has many options.
Work out before you leave your home how to get where you need to go, if you consider using public transport.
Or use this link for an app, from a comment by @Kaël.
Public transport is often busy, specially in the rush hours, but most of the time it is the fastest and cheapest way to get where you need to go.
It is certainly not living hell outside the peak hours.

Public transport tickets
Metro tickets, which you can also use for the bus and the inner city part of the RER network, are available on many of the main international trains, otherwise, you can buy them at the metro stations which are part of the train station complex, out of the machines or from a ticket office.
As far as I know the bus drivers will also sell tickets but I have not tried it.
You can also buy a pass for a few days or a pass for public transport and museums.

Bus
I would advice the bus as the best option for public transport.

• From gare du Nord there is a bus line (#42) to the Champs-Elysees, a few minutes away from your hotel. This travel does not require you to change, all things going well, and as such it is an easy way to travel.
• Bus #43 to Haussman-Courcelles, and a short walk, see answer by Gilles for details.

The busstops are just outside the train station, at the front of the station. Beware, buses do fill up and it might be better to wait for the next one if you find the bus quite full already.

Metro
Metros often are the best way to travel in Paris but from Gare du Nord to your hotel you have to change at least once and will still have to walk a few minutes, and there are several options.
Google maps (select the public transport option) will show them. Non seems easier than a bus.

• M 4, change at Chatalet to M 1, get off at George V.
• RER B or D change at Chatalet/Les Halles to M 1, (you can also use RER A and change later to M 1)
• M 5 to PLace the Italy, walk to Oberkampf and take M 9 to Saint-Philippe-du-Roule.
• M 2, which is a bit farther from the station and at its back, to Charles de Gaule-Etoile, which is still quite a walk from your hotel, so you might want to opt for a bus or M 1 to get nearer.

Each of those routes leaves you with quite a bit of walking within the metro stations, or between metro stations, and a walk at the end of the travel. I would judge it easy without luggage but not with a full set of suitcases and hand luggage.
There is a difference of opinion on whether to take the RER trains, as they are basically meant for long distance travel and can suffer from delays more than the metro system that is limited to the city. The cost is the same as long as you stay within the inner city boundaries.
If you use an InterRail or Eurail pass the RER trains might be included on the pass where the metro trains are not. (Do check before using them, please.)

Get a map
When at a metro ticket office at a main station or a tourist information office, as for the 'grand plan', the map with the metro lines, and often also the bus lines, printed over the map or Paris.
This is a very useful tool when you walk around in the city and need to take a bus or metro to get where you want to go.

Too long for a comment, but a warning and/or opportunity with taxis. On exiting the Gare du Nord, you will see a queue for taxis, which might be long (it's taken me 20 minutes in the past). If you look like a tourist, someone will come up to you and offer you a ride to 'jump the queue' - either in a 'taxi' (no idea of the licensing status) or on the back of a motorbike (ditto). This has happened literally every time I've got off the Eurostar and gone to find a cab.

I have somewhat foolishly done this once. In favour: I did actually get to the meeting I needed to get to in 30 minutes, as opposed to 50 (i.e. with a 20 minute wait), which would have left me late, and I did not die in the process. Against: I think it cost me about €50 (which wasn't an issue as I was expensing it, and it was important to attend the meeting on time). However, beware, as I guess they scalp tourists based on how much money they have; were this not the case, the locals would not be queueing; if I did this again I'd agree the fare in advance.

Regarding taxis, see Where is the Gare du Nord taxi station? Depending on where you come from/where you are in the train, you might indeed have to wait a bit.

The main alternative would be the metro. It's a bit more effort if you don't know it at all but it should be perfectly safe.

The easiest way, but not necessarily the fastest way, is to get a taxi. There's a taxi rank at the station, at times there can be a long queue. This is obviously not the cheapest way (around €15, I think, more at night or if there's a traffic jam).

By public transport, type in the address in one of the transport planners, for example RATP (the local transport authority). All public transport is well-signposted. There are line maps in every bus (except when a bus is switched from another line in a hurry) and a display with the name of the next stop (except when it isn't working). Figure out which line(s) you want to take and remember (write down) the line number, the stop where you want to get out, and the end point (all signage uses “line N to Stop Name”, there are no compass point indications). There are two main networks: the bus and the underground (métro). The metro involves staircases at pretty much every station, so it might not be best with luggage. All transport tends to be crowded at peak hour.

To go from gare du nord to your hotel, you can take bus line 43 and get off at Haussman-Courcelles, then cross boulevard Haussman and take rue de Berri until you reach your hotel (about 200m).

The metro (line 4 then line 1) to George V would be a few minutes faster, and with less risk of getting caught in traffic, but less pleasant, especially with luggage.

For public transport, get a “carnet” of 10 tickets and use them at your leisure (there's no expiration date). You can use one ticket for either one bus trip (including transfers up to a certain duration, but no return trip; tram counts as bus) or one metro trip (including RER, but RER inside Paris only). The passes for locals are good value if your stay is aligned to their validity (Monday to Sunday). The tourist passes are tourist traps.