Performing such firebox installation / reusing of the existing chimney will depend on a few factors, therefore, I would definitely verify all your installation decisions with a local professional and with the manufacturer of your new firebox. An existing chimney is one of those structures that need to be visually inspected (inside and out) periodically and before attempting to attach to it any type of fuel burning appliance.
The chimney has to comply with venting requirements of the firebox, so it has to be measured (height and flu diameter) to make sure it is not too big (it would not provide sufficient draft) or too small which could result in spillage of the exhaust gasses into your cabin. I know nothing about the firebox (model, manufacturer, fuel) or the chimney (type – interior/exterior, material – masonry/metal, dimensions, lined or not, insulated or not) so I can’t really suggest anything.
I have entire post describing all the requirements for solid fuel burning appliances venting which you find here - http://www.checkthishouse.com/2691/solid-woodcoal-fuel-burning-stove-chimney-venting.html
Installing a 90 connector may not be a problem but the connector needs to be incorporated into your chimney calculations (firebox venting requirements + existing chimney parameters) because it slows down the draft. For example, a shorter chimney could be sufficient for venting of your new firebox without a 90 connector, however installation of a 90 connector could impede exhaust gasses flow resulting in excessive condensation inside the flu as well as Carbon Monoxide spillage back into your home / CO poisoning.
Also, the horizontal section must maintain min ¼”/ft rise from appliance collar to chimney entrance.
Please let me know if you have any other questions.