What you have described has been done many times in various ways.
1st common option: As described by Focht3, soldier beams can be driven or drilled into place either under or through the existing bridge deck. Then you excavate and install lagging (and possibly tieback anchors if required) until you reach the proposed subgrade in front of the abutment. Then some type of permanent facing, either C-I-P or precast concrete, is attached to the soldier beam. You should also install a drainage medium behind the facing - either geocomposite chimney drains or free draining stone if using precast facing. Focht3 is also correct about having to shut down some traffic lanes on the bridge if installing soldier beams through the deck.
2nd common option: Working totally under the existing bridge deck, you essentially install (usually) hand-dug, C-I-P concrete soldier beams at about 8 to 10 feet on center. These soldier beams will resemble concrete underpinning piers and may be located either out in front of the abutment or they may be tucked under the abutment footing if there is sufficient space between the existing foundation piles. The soldier beams can be either reinforced concrete or you may set a steel beam into the excavated pit prior to filling the pit with concrete. After the hand-dug soldier beams are completed, you excavate in lifts of 5 feet maximum and install timber lagging between the soldier beams until you reach subgrade ot reach the level of any required tieback anchors. Drilling through the soldier beams, install and test the tiebacks if required. These will probably need to be permanent, corrosion protected anchors. After the tiebacks are installed, continue lagging in lifts down to proposed subgrade. Again, install any required wall drainage. Then install a permanent facing which is attached to the soldier beams.
3rd common option: If the soils beneath the existing abutment footing can be excavated to about 5 feet deep without collapsing, and if you can drill beneath the abutment without hitting the existing foundation piles, you may be able to install a permanent soil nail wall instead of a soldier beam wall. To build the soil nail wall, excavate in lifts of about 5 feet high, install the permanent corrosion protected soil nails and shotcrete. Install geocomposite chimney drains on the dirt before you shotcrete and connect them to a collection system to carry away any water that might appear. After you have soil nailed and shotcreted down to subgrade, you attach a C-I-P, shotcrete, or precast permanent facing. The initial shotcrete acts as the temporary facing until the permanent facing is attached. Reference: FHWA Publication No. FHWA-SA-96-069, Manual for Design & Construction Monitoring of Soil Nail Walls, November 1996.
Warning! If you have a ground water problem under the existing abutment, any of these walls will be difficult to build, especially the soil nail wall, without proper dewatering.